I co-built a similar screen sharing app (with a web server seeing traffic in the middle though) many years ago. MPEG isn't a good fit. We tried JPEG but it didn't look great. Version 1 decomposed the screen into regions, looked at which regions changed, and transmitted PNGs.

The second version tried to use the VNC approach developed years ago by AT&T and open sourced. The open source implementation glitched just a bit much for my liking. Various companies white labelled VNC in those days; not sure they fed back all their fixes. But the raw VNC protocol has a lot of good compression ideas specific to the screen sharing domain and documented in papers. People also tunneled VNC over SSH. I jerryrigged an HTTPS tunnel of sorts.

After a while I started to suspect if I wanted to get higher frame rates I should use a more modern screen sharing oriented Microsoft-based/specific codec. But it wasn't my skillet so I never actually went down that route. But I'd recommend researching using others screen-optimized lossless codecs, open or closed, so you don't reinvent the wheel on that side of things if you're serious about reducing bandwidth.