The ST lounge finder page doesn’t list anything, only a Schengen lounge for AF, KL, and RO. That doesn’t mean there is no lounge though, as they only show lounges for MU in PVG and nothing for FM. Suspect it is because they are just an affiliate member.

I looked up some trip reviews of the BUD lounges to try to find pictures of the list of airlines at the entrance but didn’t see FM listed in any for both the Platinum and Plaza Premium Lounge. They could have been old though, the BUD-PVG flight seems to have started in 2019.

Still, would suspect it’s one of the lounges in 2B, and as dcas mentioned, they’re fine. I’d consider them a significant step above the MU lounge in PVG (amenity wise, but BUD lounges are much smaller and can get crowded), unless it has gotten drastically better in the past 6 years.