We just went there yesterday, and they stated they cannot do Xrays because I don't have a PESEL.
You don't need the PESEL. The system uses national ID numbers as a replacement, and all insured people are accessible at the touch of a button to them. Ask them to search for you by name, or if not, by national ID number (that ZUS has on file).
It's a common problem as they're simply used to searching by PESEL, but since they 'turned off' the PESEL for foreigners, all government systems (with the exception of one very important one...) use the national ID number instead.
The problem is that like in most countries, medical receptionists are usually power tripping mrons who aren't willing to do a second of work to establish whether or not you have insurance. If you cannot be found on the system, then you should ask your employer for the ZUS RUMA document, which will prove your insurance status.