Xcode – ObjC app build gets build error for call to swift function within ObjC static library

made a test project to be able to replicate your errors.
You are close but you need to take care of how your static lib has its internal methods and classes exposed in its header so you can use them elsewhere.

let’s begin with the Objective-C Static Library project. Hub_lib.h

#import <Foundation/Foundation.h>

// we want the lib header as clean as possible
// it will then be imported in your project with '#import <Hub_lib/Hub_lib.h>'

// auto-generated swift -> objc bridging header imported here will
// mess it up when imported somewhere else.
// so when using swift->objc bridging place next line in .m file instead
//#import "Hub_lib-Swift.h"

//@class BLE_Central; // pre-declaration of a later fully declared Class.
// as we moved the property into .m file we dont need it here.

@interface Hub_lib : NSObject

// to make this work you would need pre-declaration of BLE_Central, see above interface
//@property BLE_Central *ble; // placed in .m interface extension instead.

-(void)run_central;

@end

static lib counterpart / implementation Hub_lib.m

#import "Hub_lib.h"
#import "Hub_lib-Swift.h"

@interface Hub_lib ()
@property BLE_Central *ble_central;
@end;

@implementation Hub_lib

-(instancetype)init {
    if (!(self=[super init])) return nil;
    _ble_central = [[BLE_Central alloc] init];
    return self;
}
-(void)run_central {
    [_ble_central run_central];
}

@end

notice BLE_Central property is placed in the class interface extension and when you want to use swift module stuff that exposes back to objc you need to declare the auto-generated bridge somewhere (best done in .m file #import "Hub_lib-Swift.h")

your BLE_central.swift with its protocol method implementation

import Foundation

import UIKit
import CoreBluetooth
import os

var centralManager: CBCentralManager = CBCentralManager()

class BLE_Central: NSObject, CBCentralManagerDelegate, CBPeripheralDelegate {
    
    func centralManagerDidUpdateState(_ central: CBCentralManager) {
        os_log("centralManagerDidUpdateState")
    }
    
    var discoveredPeripheral: CBPeripheral?
    var transferCharacteristic: CBCharacteristic?
    var writeIterationsComplete = 0
    var connectionIterationsComplete = 0

    let defaultIterations = 5  // change this value based on test usecase

    var data = Data()

    // as you figured out before we need to expose to @objc
    @objc public func run_central()
    {
        os_log("run_central")
        // out-commented for testing purposes
        //mobile_sys_hub_lib.centralManager = CBCentralManager(delegate: self, queue: nil, options: [CBCentralManagerOptionShowPowerAlertKey: true])
        os_log("Scanning started")
    }   
}

As long no extra code from objc is used in swift inside the static lib, the
Hub_lib-Bridging-Header.h is empty

//
//  Use this file to import your target's public headers that you would like to expose to Swift.
//

Next let’s see how to import your static lib in your "sim backend UI" Objective-C Project app.

Goto "sim backend UI" App Target settings > General > Framework, Libraries, and.. > hit the + button and search for your compiled libHub_lib.a file. > select it > hit ok. Should be very much in your framework list by now.

Yes, thats not enough! You have to declare its header in your app project somewhere. Where exactly is up to you. Instead of implementing it multiple times we do the following in BLE.h

#import <Foundation/Foundation.h>

#import <CoreBluetooth/CoreBluetooth.h>
//#import "Hub_lib-Swift.h" // no no no no
#import <Hub_lib/Hub_lib.h> // much better. test-compile once if it is crying

NS_ASSUME_NONNULL_BEGIN

@interface BLE: NSObject
// <CBPeripheralManagerDelegate> // "//<Protocol>" will confuse doxygen

@property (strong, nonatomic) CBPeripheralManager* peripheralManager;
@property (strong, nonatomic) CBMutableCharacteristic* transferCharacteristic;
@property (strong, nonatomic) NSData* dataToSend;
@property (nonatomic, readwrite) NSInteger sendDataIndex;

-(void)start_central;

@end /* BLE_h */

NS_ASSUME_NONNULL_END

counterpart BLE.m

#import "BLE.h"

@implementation BLE

-(void)start_central
{    
    NSLog(@"invoked BLE start_central");
    Hub_lib *Hub_central = [Hub_lib new];
    [Hub_central run_central];
}

@end

Your ViewController.m or .mm makes use of BLE.h with its already imported static lib header

#import "ViewController.h"

#import "BLE.h"

@interface ViewController ()
@property (nonatomic) BLE *ble;
@end

@implementation ViewController

- (void)viewDidLoad {
    [super viewDidLoad];
    // Do any additional setup after loading the view.
    
    UIButton *btn = [UIButton buttonWithType:UIButtonTypeSystem];
    btn.frame = CGRectMake(100, 100, 200, 50);
    [btn setTitle:@"run simple central" forState:(UIControlStateNormal)];
    [btn setTitleColor:UIColor.greenColor forState:(UIControlStateNormal)];
    btn.layer.backgroundColor = UIColor.orangeColor.CGColor;
    btn.layer.cornerRadius = 5.0f;
    [self.view addSubview:btn];
    
    [btn addTarget:self action:@selector(run_simple_central:) forControlEvents:(UIControlEventTouchUpInside)];
}

- (IBAction)run_simple_central:(id)sender {

    // BLE needs to be allocated to take effect.
    if (!_ble) _ble = [[BLE alloc] init];
    
    // testing.. should log "run_central" + "Scanning started"
    [self.ble start_central];
}

@end

Made a testButton so you can see if the implementation invokes what you expect. Compile!

And? Aoutsch! Does not work. What happened?

If your App is a plain Objective-C project it doesn’t know about swift yet and will complain in a weirdo way, possibly via something like

Link: Could not find or use auto-linked library ‘swiftCoreImage’
Undefined symbol: value witness table for Builtin.UnknownObject*

Solution: Most easy way is to create a swift file in your app project and allow Xcode to make a bridging header for you. (Alternatively change project settings, search for "bridge" and go step by step thru the properties)
The swift file does not have to have special content.

//
//  MakeProjectSwiftCompatible.swift
//
import Foundation

Compile again. now it should work because the partly implemented swift module from within your static lib can properly work when your Objc-App-Project is able to work with swift stuff.

Edit as you asked to work with Objective-C++/C++ in your static library things change a little bit.. so here some additional example code to proof it.
In Hub_lib project (targeting your "framework") add some files which will keep some random testing c++ code

//HubCPP.h
#import <Foundation/Foundation.h>

#include <vector>

NS_ASSUME_NONNULL_BEGIN
class SomeCPPClass
{
public:
    SomeCPPClass(id<NSObject> obj, size_t size);
    ~SomeCPPClass();
    id<NSObject>                    getBuffer() { return buffers[bufferIdx]; }
    unsigned int                    getCurrentIdx();
private:
    std::vector <id <NSObject>>     buffers;
    unsigned int                    bufferIdx;
    bool                            isReady;
};
NS_ASSUME_NONNULL_END

To make Xcode know that you work with C++ you need to have implementation file ending with .mm and also need to change HubCpp.h "Identity and Type (right Xcode panel while file selected)" to C++ Header

// HubCpp.mm
#import "HubCPP.h"

SomeCPPClass::SomeCPPClass (id<NSObject> obj, size_t size) :
bufferIdx (0),
isReady(false)
{
    uint8_t ringSize = 255;
    assert (ringSize > 0);
    for (uint8_t i = 0; i < ringSize; i++)
    {
        //buffers.push_back ();
        bufferIdx = (unsigned int)size;
    }
}
SomeCPPClass::~SomeCPPClass() {
    // cleanup allocated stuff here.
}

unsigned int SomeCPPClass::getCurrentIdx() {
    return bufferIdx;
}

Rename Hub_lib.m to .mm and change its import rules accordingly to the following ..

#import "Hub_lib.h"
#import <CoreBluetooth/CoreBluetooth.h> //needed because the -Swift.h bridge will cry in the next line
#import "Hub_lib-Swift.h"
#import "HubCPP.h"

lets change the proofing method in Hub_lib.mm so it really uses C++

-(void)run_central {
    [_ble_central run_central];
    SomeCPPClass *cpp = new SomeCPPClass(@"justSomeNSStringObject",2);
    unsigned int idx = cpp->getCurrentIdx();
    NSLog(@"objectiveCplusplus testIdx = %u", idx);
}

Compile Hub_lib (scheme). It should work by now and also accept the use of #import <vector>.

If this works go on and change your Objc-Project-App.
Switch your compile Scheme to target your Objc-App.
Change file name BLE.m to BLE.mm (makes it a Objective C++ Source)
Change file name BLE.h to BLE.hh (makes it a C++ header)
Change in BLE.mm #import "BLE.h to #import "BLE.hh
in ViewController.m kick out the line #import "BLE.h" and replace it into ViewController.h instead as #import "BLE.hh"
(In general its much easier to keep your compiler informed what language to expect in implementation when you place import headers in header files.)

Compile. Thats it! Your Objective-C++ static lib should properly work at this point.

Edit
You can find a ready made workspace for Xcode here…
github: combine cpp swift and objective-c in static lib