FLEX or NYLON I would guess

Need to know what material your build plate is made of to guess what wouldn't stick to it.

TPU if it is soft and like gummy/rubbery

I use the stuff for seals some times, but more often it makes the perfect string for cat toys. Put a length on a stick, add some painter's tape on the end, and wave it around; cats go nuts.

Well, if we're ruling out PET and PETG -- PET would probably require the temperatures you describe and would be impervious to acetone, as well as extremely flexible -- there is an outside possibility it could be HDPE.

HDPE filament is damn rare, though, and I'd doubt anyone would be giving it away as samples given how difficult it'd be to print with most consumer machines. HDPE's signature tell is that it feels somewhat waxy if you e.g. scrape at it with your fingernail.

Well, I toyed with the printer's temperature settings until I found a combination of nozzle temperature and bed temperature that finally made it spew out parts without making a disaster.

It seems to flow best at 260C, but then when it lands on the bed, it immediately shrinks as it cools and the part curls up and comes unstuck. I had to lower the temperature to 245C and raise the bed temperature to the maximum this printer does - 110C - for the part to stick enough to complete.

The nozzle clearly isn't hot enough because some layers on the final parts look like they're about to delaminate. But any hotter than that and the material curls up. And it doesn't matter what bed plate I use: that stuff doesn't seem to stick to anything properly.

I printed spectacles with that mystery filament. You can see how floppy it is here:

https://toobnix.org/w/qJJ1htb9eqmiHx7gSpq2RT

It looks like TPU alright, but the temperatures involved aren't really typical of TPU. Also, acetone does nothing to it whatsoever.

The material also doesn't like to be filed or sanded, and the best results for a nice finish without bits of material sticking out all over the place is to "polish" it by running a very sharp x-acto blade across the surface until all the junk is gone and the surface is shiny.

Weird filament. Kind of useless...

Sounds like TPU? Maybe soft one too.