Ouch… I admittedly don’t know much about Supabase, but odds are that your current machine would need to be running in order to migrate, 
…
It might be possible to override the existing machine’s startup sequence, so that you could at least SFTP the raw data files off of it.
(Caveat: the comment about fly pg import in that older post is unfortunately over-optimistic; too late to edit!)
It would still be necessary to feed that to a Linux Postgres v14.x instance running locally (on your desktop/laptop), 
, which takes a lot of unfamiliar steps if you’ve never run PG that way before.
Alternatively, you could try contacting billing@fly.io and see if they might lift the restriction long enough for you to grab† your data. Any user can send mail to that address; it’s not restricted to the paid plans the way other support is. There may be a few days’ delay before a response arrives, though…
†If the PG server is running, then pg_dump is the best bet. (Or pg_dumpall, if you were doing fancy things with roles, multiple databases, etc.)