I think there might be a way with something like virtual audio cable? Have the screen reader output to a different device from the audio you want to stream, even if that means virtual shenanigans.
Other than that, the best I've got, OTTOMH, is to use a braille display and mute speech, if that's a viable option.

@CrystalD,
It really depends on how much you're willing to spend in the money department.  VAC is a wonderful and convenient little option I myself don't use because I have an audio interface/10 input stereo mixer and can rout audio in ways I never was able to before now, with just a few button pushes and knob twists.  I can monitor things without having them heard, have people hear things I myself don't want to hear, which is rather useful when someone on my TT server is enjoying something I myself am not interested in or have already heard, and beyond that, spice up my audio life with audio processing and a bunch of other bells and whistles you probably don't need.  The options are there though, so the long and short of it is, yes, there is a way, a myriad of ways, in fact, to do what you wish, and we're here to help you, based on what you choose to do.

I have a audio Interface mixer but I see now way to rout programs to it. Its a two input output mixer. Is this only possible on digital mixers. I would really love to do routing from the mixer level.

Do you have to have wires or cables or is there software it comes with.