So this other lock also looks to be LF also
I’m making a presumption here, but I would be 95%+ sure thats why the reader has LF printed on it.
I will need to google to confirm, but I believe PAC is pretty odd, in the fact that it does the typical / usual 125kHz, but I believe it also does a more unusual 150kHz …153 I think???
Anyway, I don’t have one of these PAC fobs/cards to try, so I don’t know if / how you can clone that to a T5577 chip.
Even if you could, how well it would work.
eg. T5577 is usually used with 125kHz but will also work with 134kHz (animal chips), so a ~150kHZ may be a bit too far off, but 
Nope, the fact that both readers are LF, that is why TagInfo can’t “see” the fobs, as TagInfo ( your phone ) is only capale of reading HF.
So thats good-ish news
LF is almost always super easy to clone.
BUT
As I mentioned above, with yours, there are a couple of unknowns ( to me at least )
However
since you have 2 tags, and BOTH appear to be LF, you would need 2 T5577 chips.
I know the T5577 can hold more than one EM mode ID, but I don’t think / know if it can hold other modes.
All that being said, I have a few more questions for you
Does either fob open both doors?
Do you have a Proxmark3? or similar RFID Diagnostic tools? Flipper / iCopy etc.
Could approaching the access admins be a possibility for you?
If yes, then, your easiest would be for them to enroll your ring to both systems.
But you still have 1-2 problems.
You’ll need to change your ring from EM mode to Prox, and you’ll need a PM3 or similar to do that, and also, the unknown info around the PAC reader and the mode it is using, Hopefully, it is the 125kHz not 150, and it is also compatiable with Prox.
I hope that makes sense