I guess this mainly a question for the Teams experts. My experience of sharing individual windows with Teams is so-so. Sometimes it works, others it does not. I have not thought that much on whether it is SSMS or something else.
When I present over Teams remotely, I connect a big monitor to my laptop where I have Teams etc. Then I share my laptop screen. (Where I have a VM from which I run the presentation.) I think you should consider some similar arrangement.