Start with the charging circuit before getting your hands dirty in the carbs (not that what you are planning in there will do any harm).

First cause is the flat battery, checking the charging circuit will tell you if it was faulty or not. Hondas reg/rec units from this era are known to be a bit dodgy (ask anyone with a honda 400 - I went through 3 or 4 personally).

If the battery is simply co-incidence you might also have an issue elsewhere - but I very much doubt it.

If you want to go all gucci on the battery, get a gel one.