More new info!.. I first read "satellite modem" to mean it was secondary to another modem. My bad! Not many such systems in my part of the world.
Yes, very likely that satellite latency or varying bitrate or short gaps can cause the problems. As I understand it the handover between passing satellites can be messy. Especially a problem if it's rainy, or even if the air is damp, or if there are electrical storms.
Nevertheless, mesh systems can also perform badly for real-time traffic (or more precisely, time-sensitive traffic) - kit varies, some is ok some is not so good. Maybe the two things together are a 'perfect storm'?
A test to see if the LAN part itself has problems, might be to plug a laptop into the satellite modem unit using ethernet only, running up a little DLNA server on it with some locally stored files (24/192 files if possible - it's a tougher test). Then set your media streamer to use that PC server to get your files. If they play perfectly over the longest reach of the mesh, especially if there is other traffic going on, then you can rule the mesh out.
If you have a reasonable mobile phone signal, you might be able to the phone as as WiFi hotspot, also as a test. Link your streamer to the hotspot and see if the audio is OK from Qobuz or whatever you use.