If it's any help, I had to have a GA for a short repair operation after DS born, when he was 12 weeks - he was exclusively BF and I mean BREASTfed - wouldn't take a bottle, cup or anything else, so the hospital had to give me a side room and let him come with me for the day (baby friendly hospital so I threatened to kick up a stink!), my mum had him while I was in theatre/asleep. Anaesthetist reassured me that he'd be fine and he was - I probably fed him about 2 hrs after coming round and he didn't seem affected at all.
You can have other painkillers while BF, even ibuprofen, and can combine this with paracetamol for a couple of doses if pain really terrible, though I would recommend sticking to paracetamol if you can.
I had 3 GAs in the 6 months after dd2 was born (complications of pregnancy getting fixed) and always fed as soon as I got back to the ward. GA drugs are very short acting and are reversed by the anesthetist who brings you round, so once you are awake they are gone.
Hope it all goes well for you.
(I ended up not seeing the HV - you know you don't have to after the first visit if you don't want to, you just have to say "thanks, I'll call you if I need you" - after about 5 visits with my dc1. She was totally useless)