Hi Jakub,
As a fellow Pixel owner (I started with the 2XL), I keep an eye out for Pixel issues and tend to remember them. I've replaced the battery on my niece's 4XL with no issues and I can tell you that there haven't been any posts in the last couple of years describing any issue with parts pairing like you're talking about.
In fact, I'm not aware of any parts pairing done on Pixels; even the fingerprint sensor can be replaced, unlike the home button/Touch ID sensor on an iPhone.
The upshot here is that replacing the battery should resolve any performance issues without having to go to Google for some kind of unblocking. If you add to that the fact that iFixit sells genuine Google parts, you should feel safe in ordering a battery from them and using their guide to replace it.
[product|IF356-271|Pixel 4a Battery: Genuine Part / Replacement Kit - iFixit]
[guide|139563|Google Pixel 4a Battery Replacement - iFixit Repair Guide]