On what device are you running iReal Pro?
I have an iPad Mini with Retina display (now called the iPad Mini 2) and a Bose Soundlink III bluetooth speaker (which, by the way, does sound very good with iReal Pro and everything else). I have always noted a slight delay between the presumed first beat of iReal Pro playback (by my own counting in along with the iReal Pro screen) and the sound coming out of the speaker. This occurs with all of my bluetooth devices; I regularly use a Logitech bluetooth adapter for iReal Pro audio output. When I read this post, I did a few informal measurements.
I paired my iPad Mini with the Bose speaker and cranked up iReal Pro using a song played back at 150 beats per minute. Since one minute contains sixty seconds or 60,000 milliseconds, beats at this tempo occur 400 ms apart; sixteenth notes occur 100 ms apart. I found that the delay was of the order of a sixteenth note, or about 100 ms. This is a very rough estimate, of course. I did the same with the Logitech and got the same result. In neither case is there any noticeable delay between the playback locator on the iReal Pro screen and audio output.
I note that you have the same problem with a wired speaker. This seems to indicate that the problem is with playback (in general) on your device.
Two questions:
Is there a sync problem between your device when it plays a video (either pre-loaded on your device or streamed) and the Bose speaker via Bluetooth? If so, the problem is not with iReal Pro. Likewise, when you link your device to the wired speaker, is there a sync problem when playing a video or iReal Pro? This would indicate a general problem with audio/video syncing on your device.
I've had success linking my iPad Mini with bluetooth devices. I hope we can discover a solution to your problem.