OKay, so I just checked the Compatibility Matrix Sheet, and the Paxton P38 system on there has been confirmed with the Mifare Classic 1k, from this link
not to say it is not compatible with other systems, It just hasn’t been confirmed yet ( please let us know if you manage some more success)
It looks like I was on the right track with the Paxton system, I found this
From this document
Paxton.pdf (182.1 KB)
(I have also put it in the .pdf repository)
With regards to the Paxton, it looks like you have a few options.
BUT
I also found this, which looks like you have to activate it with a licence in the form of an access card…
https://www.paxton-access.com/ae/125khz-iso-proximity-card-licence-x-1-with-genuine-hid-technology/