Recording your synth is the correct solution
I'm fairly certain you can do this in BiaB

Though to be honest I'd be a lot more qualified to explain how to do that in Realband (the sister program you probably have if you are using Windows as OS)
I find RB easier, but I'm pretty sure you can do this in BiaB using the routing options .. or the routing options of your interface

Example:
Send the audio out of the external synth to channels 5/6, then record the input from 3/4 after routing those inputs (5/6) back in on 3/4 ..

or fix the problem with the Realtek StereoMix mix (in my opinion, not the best solution, but usable)
Driver update maybe?
Or configuration inside the Realtek (?)

I do not work here, but the benefits are still awesome
Make your sound your own!