Okay.
Here are my thoughts.
Because both cards didn’t read on TagInfo, and they both allow you access, they must be LF only.
Because both cards are labelled HID I am going to assume they are HID Prox.
Which your bluecloner can read/write from/to.
It is strange however that your LF LED on the Diagnostic card is not illuminating
and also that you could only successfully write the site B card.
Because you have better access to the Site A system, I would suggest:
Waiting for your Proxmark then:
Remove the 
password
Write the Site B to your NExT ( actually to a test card/fob first )
then enroll that same info to your Site A profile on that system.
This in theory will give you access to both sites with one card / Implant
In fact, in the interim I would grab a test card T5577
With your , copy and write the Site B info to your card.
Test that at Site B
If that works, try to enroll that card at site A.
If that works, the only thing left to do write the site B info to your NExT, then test the coupling of your NExT on the readers.
I know you didn’t manage to write Site B to your NExT previously, but that can sometimes be fickle.
If you can’t write to the T5577 card, then we may need to wait for the Proxmark3 
to do some more diagnostics.
This may help with your NExT read / writing