I had sleep apnea and got jaw surgery. But I also have a dysfunctional TMJ, so my mouth opens when I sleep. Every chin strap on the market pushes the chin down into the airway, it seems. Which one did you get?
But, if you can breath through your nose well enough, I think some kind of mouth tape can work? I haven't tried the Hostage brand anti-snore tape, but maybe someone here has had experience with that.
If you really want the chinstrap that I got, I got this one: https://www.amazon.com/gp/product/B08DR16XGV/
Based on the pictures, it obviously pushes the jaw backwards. That obviously wouldn't work for you. I wear mine in such a way that it is helping my jaw close and stay shut. I agree that you don't want something pulling your lower jaw backwards.
Though I have this paranoia that mouth tape will increase my philtrum length, which I know is a vain worry. Another solution is one of those incline pillows - just a 20% gradient should allow me to back sleep, which I prefer.
Post-surgery I had my jaws wired shut and they had this chin strap that was like an ace bandage that around my head and chin. I could replicate that with one of those mouth guard combined with a chin strap, but I read somewhere that this would eventually pull the upper jaw down towards the airway as well (since maxillary plates are not fixed.)