Hi,
Assuming your existing battery is fine, a bigger battery alone would not help, because as soon as the engine is started, the system voltage goes higher than the battery terminal voltage....12.5V to 14.5V (ish). So, once the engine is started, the source of all current comes from the alternator.
If you’re convinced that the alternator is at fault then the standard one should be fine if either repaired/reconditioned or replaced.
A bigger battery will allow you to run electrical without the engine running for a little longer. Or will allow you to draw larger currents from the battery while the engine is off (for starting and electricals)
A bigger alternator will allow supply more juice (current) to the devices in your car, but I’m not sure that the existing wiring will be rated high enough to carry higher currents than the standard alternator can supply (melt…smoke…fire?).
I’ve been thinking about the slow idle thing and I must admit I'm pretty stumped!
You could eliminate the alternator if you have a mate with a K10 that will let you borrow theirs (known to be in good order…..fast idle is working on their car) to try....better than shelling out just yet?
The only other thin I can think of is that battery might be dicky? Does your battery get hot after 5 or 10 min of idling?
Perhaps you could borrow a mates battery as well to test with?