Hey everyone, I need your technical help

I bought my first Crossfire 3 weeks ago, amazing car, loved it since minute one.. However the central locking is not being too friendly with me.

If the car is stationary for a while, the central locking works with no problem. But when I park the car after a ride or turn the key to the second position, the car refuses to lock with key.
I found out, that it takes approximately 40 minutes until it lets me use the key locking again. Also, locking with the button inside the car works everytime with zero problems.

I replaced the central locking pump yesterday for a used one, but nothing changed:')
Batteries in the key fob has been replaced for a fresh pair.

Also- the alarm has been most likely tampered with (somehow plugged off or rusted), as it doesn't work. I actually don't mind it not working, but I'm thinking it could be a clue..
Some technical stuff- 2004 coupe, 155k km (96k miles), central locking pump replaced, two button keyfob

Thank you for taking your time to help me
(English is not my first language, please be patient with me:-) )