BMW E39 523i Struggling to Start: Low Key Battery, Voltage Issues, and INPA Error Checks

Hello.
I have the following problem:
Every day I do about 50km from / to work and never (even when the weather is cold) there was no problem with firing the car. A message about a low battery in the key appeared on the board a week ago. I ordered 2pcs cr1220 Duracella. I exchanged in less than 60 seconds. I fired a car twice on the same day. The next day the car was standing all day. The next day he did not fire anymore (he spins but does not fire). The voltage on the ACU 11.6V (battery 4 years centers futura 77 Ah). For 12 hours I charged up with cheap charger hb1206s (I know that I had a bad but no better one). The charging voltage increased to 14.97V. After putting the car into the linden again. Borrowing electricity from another car - lime. New battery - also does not start. The old inpa does not show any errors (in the engine and body-ignition) unless you do not look where you need - I ask for directions where to look (coding option did not move).
I have 2 keys currently in use and the second one with discharged batteries - a battery - it does not fire on both.
Is this the immobilizer fault? Maybe a fuel pump?
Are the batteries from the remote control also responsible for the immobilizer?
I did not repair the erasers from the buttons so I could not damage / lose the transponder in the remote.
How to diagnose it?
No new controls appear.
Fuses in the glove compartment.


please help

First let the colleague synchronize the EWS DME module using the INPA program. The chance of disarming the EWS module (module responsible for securing the vehicle against theft) is small, however, it can not be excluded.
If the synchronization of the modules described above does not change anything, then one of the more likely causes of this fault may be damage to the crankshaft position sensor or the crankshaft position sensor. When no signal from any of these sensors is supplied to the motor controller, the motor can not be started. It often happens that the sensors themselves are functional, and only the insulation of the signal lines is damaged, which leads to their short-circuiting. Therefore, it is worth checking the condition of the cable insulation first, in particular, under the rubber cover located next to the sensors themselves.
Moreover, let the colleague check if the diagnostic interface connects correctly with the engine control unit.

Buddy had an accident in the BMW e39 520i that he did not want to run because of the fuel filter, maybe start from this.