For whatever it’s worth, newer Intel wifi/bluetooth chipsets (e.g. AX200 or AX210) have consistently worked very well for me under Linux. If your EliteBook has a socketed wifi/BT card, it might be worth buying an Intel card (they’re dirt cheap) and installing it.
My issue with battery life isn’t getting it to match what it is under Windows, but that x86 ultraportables tend to have disappointing battery life under either OS. Linux doesn’t make it worse but doesn’t fix it either.