I'm pretty sure that screw is the problem. I have a few held by similar set screw.
Most unscrew in regular direction, I just remembered one turns in opposite direction, have you tried that?
The picture looks sorta in between regular Allen and Torx. Sides of opening kinda rounded, the corners not as acute as torx. It may just be worn from lots of removing and/or overtightening. If torx don't fit, t least you have drivers, seems like everything from cars, appliances to decks now has a torx somewhere on it.
I've found that if you have a wide enough assortment of Allen wrenches including metric, in small enough increments, some old and worn to a non specific size, some filed down in the past to fit some odd ball size. Some metrics fit some regulars and vice versa. Some will loosen some torx if not too tight.
If you have a replacement lock set, and drilling doesn't release old one. There's always hack saws, and sledge hammers.
You might take pic of that screw to a locksmith, if its proprietary they might recognize it and have tool. Might be a common set in your area and locksmith is familiar with it. Have you talked to land lord about it?