Not all that familiar with your lock but this is what I'd try.
Open door and manually trip the locking claw with a screwdriver. Should click twice to fully lock.
Pull outer handle and it should unlock.
Trip the lock again then lock doors with fob.
Try outer handle and it should stay locked.
Unlock with fob and feel/listen for soleniod.
If you feel/hear it then it's getting it's unlock signal.
There could be high resistance in the wiring or a connector but you might be chasing ghosts looking for it.
More likely to be weak solenoid, sticking or broken catch.
Would think it's best to remove it if you want to lubricate properly as it all looks one piece from the pics on
Complete used locks seem around £15 so I'd just be tempted to buy one so you only need to do job once.
Test the new one out of the door if possible to confirm it works.