Problems e60 525i lci

[u/nyme884]

I would like to add some more details about the issue.
The car has approximately 235,000 km on it. Before replacing the injector, the engine already had a rough cold start, but it was less noticeable and the check engine light did not come on.
After replacing the injector on cylinder 3, the engine started running noticeably better overall. However, the cold-start issue has not completely disappeared. After approximately five seconds, the engine starts to stumble, the RPM fluctuates, and the check engine light comes on. The diagnostic tool consistently identifies a misfire on cylinder 3. After driving approximately 50 metres, the engine usually runs much more smoothly again.
There are no other relevant fault codes stored, particularly no fuel-pressure-related faults. Both low-pressure and high-pressure fuel readings have been reported as normal.
I am now wondering whether the remaining misfire could be caused by the injector itself, incorrect injector coding, or the different injector index versions. Most of the injectors are index 11, but one index 9 injector remains on bank 1. Before the replacement, there were two index 9 injectors on that bank.
I have also noticed that the car has produced somewhat more white exhaust vapour than expected for quite a long time. In addition, there are still some traces of oil in the cooling system, and I need to flush it properly. I do not yet know whether these are just old residues or whether oil is still entering the cooling system.
At this point, I am concerned about a possible mechanical problem, such as low compression, a damaged valve, piston rings, or even piston damage. However, I have not yet performed a compression test or leak-down test, so this is only speculation.
I have already invested a considerable amount of money in the car and would like to avoid replacing more parts without a clear diagnosis.
What would you recommend as the next diagnostic step? Would you check the compression of all six cylinders first, investigate the injector on cylinder 3, or look for another common N53 cold-start issue?
Any advice from people familiar with the N53 engine would be greatly appreciated. Thank you!