Intelligent key can not be detected

Firstly, It's not the battery.


One of my remote batteries was dying. It stopped working really fast (too fast, a day or two).
Replaced the battery (2.78V) with a new one, 3.2V. Now buttons are working fine but key can't be
detected. I have to click the remote to open the doors and push the start button with the key.
Second remote is working fine.

That's really weird
I checked the electronics inside, they look fine, no water damage.

Any ideas?
Is there any reintroduction mode that should be done (there is on BMW definitely)?

On my VW's the key detection is by RFID and the power required comes from a coil on the dash. Is there any chance something internal to the key got shifted during disassembly/reassembly and is blocking the RFID chip?

I reviewed the service manual, but didn't see anything that jumped out.

Since one key works and the other doesn't, it would seem to point to the non-working key and not something in the car like an antenna or the receiving module.

Can you remember if the bad key ever worked at some point?

Everything was fine 3 days ago. And then just didn't open the doors. Unlocked manually with button on the remote.
No start. Tried 5-10 times with randomly clicking remote inbetween, and it worked (without bringing remote near start button).
I think buttons also did not work with old battery. Now they do - reliably and at great distance.

Measured the winding around the perimeter (thin wire). Near 100 ohms. Therefore not broken :|
What the heck is going on.
My best bet is that something happened with synchronization (rolling code) when battery struggled.