Your battery should be fine. Bear in mind that the much later Saxons, with a decent alternator, had a 5.5 Ah battery as standard. I have fitted 9 Ah, but the battery is not a problem given a good generator. My bikes do not need LED bulbs to maintain good charging.
As for the surging, it seems to be a carburettor problem as much as anything else. Turn the tickover down as low as you can get away with. Some folks set it to the minimum, which means the bike won’t tick over, and you just have to keep your hand on the twist grip when stopped. On my Saxon it is very low – the tacho pointer doesn’t move off the bottom stop which should mean it’s lower than 500 rpm. (The handbook says 1250 revs, but I can’t see why.) You may be able to get a better carb. Some of the German suppliers offer a modern Bing carb set up for older MZs, but the cheapo Chinese copies of “better” carbs are generally reported to be rubbish. It seems to me that the cure is a good carb and electronic ignition, allowing a low tickover.
When I had my 250/1 I just lived with it. The other problem can be bogging down if you try to bang open the throttle while below 3000 rpm. On the hill I used to live on in Wales that meant I had to grind to a stop and start all over again. If I hit the bottom of the hill at just over 3000 revs I could sail up it.