There are lots of solvents one could try. I agree about avoiding acetone or any ketone. The plastic on the meter would almost certainly be ruined.
Household acrylic floor stripper (not just daily cleaning stuff) might work. The one I use is made by ZEP. It contains butyl cellosolve.
I would leave the spots, and if you remove them from the panel, leave the meter alone. There is no way to predict the long term effect on that unknown plastic. You may get immediate clouding or long term crazing. If you insist on attacking the meter, I would try a little IPA (which you said didn't work) or VM&P naphtha, which is a low boiling version of mineral spirits with slightly better solvent properties. Automotive paint shops may sell something called "final prep" that is similar but more expensive. If you try the hydrocarbon solvent, be ready to wipe it off with a soft rag, if it doesn't work. It can cause crazing later. On CD "jewel" cases, those solvents cause slight clouding almost immediately. Bottom line, leave the meter alone or be ready to buy a replacement from HP.
EDIT: Buying from HP/Agilent/Keysight may be difficult. See current thread about Keysight only selling to commercial users.