Rough idle issue when hot

Ok so this is the new issue.

E46 2001 325I touring
280 000km on the clocks
Full service history

I hve searched but found no symptoms specifically related to mine

Previously car wud generally idle low and rough. Found massive vaccum leak in brake booster. Replaced it and now have new issue

Car will idle rough and low once hot.
Not immediatly but the revs will gradually drop and then the car wud misfire

Cold start idle is 100% ok. Driving the car is 100% ok. Jst at idle she goes all sick. No suction when fuel cap is opened, checked all vaccum hoses and intake boots,new plugs, oil, oil filter, fuel filter, new oil housing gasket, new valve cover gasket.

I must mention, I bought this car smashed, it started and drove still but needed a new radiator and condensor and all front panels.

Using a Delphi, picked up misfire on cyl 2,3 & 4.

No clue what to check for further.
Other sites recommend battery issue. I must admit, the battery that was in this car is a 640 nd recommended minimum battery is a 652. Also, the battery is old so needs replacing anyway, kick starting the car at times that the battery is dead

Is it possible that a weak battery can cause this?
That when kick started, she idles 100% ok, driving is 100% ok but when u come to a stop, she decides to drop idle to 500rpm and twitch and make the "clicking"noise when u punch the throttle like she wants to stall

Im replcing the battery tmrw anyway, jst wnt to knw if anyone experianced this before

No engine light, no other error codes with any sensors etc etc..
Only other light that comes on is yellow oil level sensor but I thnk that is busted cause oil level is 100%. No oil leaks, no water leaks no nothing

Is it true that low voltage on battry can cause fuel pump to pump slower? Hence dropping fuel pressure?

Also, this car was standing for 6months befoee I bought it so I thru wynns injector cleaner in, jst incase
Also, when I bought it,they started the car up for.me using a 100% good battery and idle idled for a while and it was fine, hence I bought the car, battery they gave with the car was only good enough to power up the dash lights and cabin lights etc. Wud not swing

Also, the radiator fan stays on permenantly. Its an original fan I bought off a junkyard beemer with blown headgasket due to thermostat failure and is near mint. Figured its maybe jst a relay issue but could that cause it? Temp reading is half always and stays there as per normal. does not drop or increase

Thanks in advance

Rough idle issue when hot