Okay, so I checked the spreadsheet, and NORMALLY I would put a note as to who/ where in the forum ir was tested.
There was an Amazom link but no note.
I see it was listed in an old post of Amals
And it was listed under working with xEM in 2016. Theock manufacturer may have made some changes (likely power saving due to your experience)
OR
another thought, when did you install your NEXT? could it be a swelling issue
anyway, to answer your question
So they would both be compatiable (although the FlexNExT is currently still under development), whether or not they would work, that would require some testing.
The FlexEM has a much better performance over the NExT /xEM so the odds of that working would be much better.
Here’s what I would do
Use my xFD 
keyring to test coupling and antenna placement.
Then if required, remove the cover to expose the antenna and see if possible to move the antenna closer to the surface.
If I couldn’t get the NExT to work, I would consider getting a FlexEM (which I can’t guarantee woukd work, but personally think it has a good chance of working.)
But you might want to do a test first.
Did your Proxmark3 include one of these?
The insides are pretty close to a FlexEM (20mm including encapsulation) so likely a ~15mm antenna
so try to enroll that or buy one if you don’t have one.
You can also get them “naked”
search for
125kHz Keyring
and
125kHz 15mm
which should yield you some results
Just make sure the ones you get are T5577.
excuse I am not sure what mode your Lock uses I ASS-U-ME EM41xx but easier just to get the T5577 since you have a proxmark to do any changes you need.
I hope that both helps and makes sense