Hello dmagerl, and welcome to our community!
To answer your logs question: I think you can access your Raspberry Shake rs.local/ page and, from there, tap on the orange button at the bottom to download a single zipped file with all the log files in it.
You should be able to upload that single file here. If not, feel free to upload it on Dropbox/Mediafire, or other providers, and give us the link.
Regarding the Starlink connection, other users have had some fixable issues in the past. You can check this topic, or this more general search for more information.