I had an issue similar, the remote locking and the immobiliser are two separate issues.
the dead battery in the fob won’t affect the immobiliser which is read by a benign chip in the key by the ring on the ignition barrel.
the lights on the dash should illuminate when the key is turned regardless of the engine starting.
if it doesn’t turn the dash on, I would look toward the sensor on the barrel.
do the interior lights and seats move when you get in? Is there power through the car?
when you get the car plugged into a reader, put the live data feed on and check the ews/ dme module.
when the key is inserted into the barrel, the chip should be read and the key number displayed, I think the check boxes turn green too, then the car will start. If you are getting no green lights on the programme or key numbers come up, then it’s either the chip in the fob or the chip reader on the ignition barrel.
I found this out the hard way after replacing everything including the ews and dme modules, locks, barrels and sensors…. Turned out to be a chip not being recognised, new key, new chip, programmed up and presto!
hope this points you in the right direction