I was thinking this myself, im running a 20ah golf cart battery that has started my rx7 turbo with no problems and now is on my supercharged 1.8 lotus(so harder to turn over that yours) with no issues.

Mine weighs 5.3kg, have you weighed yours? what make etc is it? only way lighter I think would be a lithuim ion battery to take it under 2kg.

Its two options that i think is responsible for your problems, either alternator is on the way out or you have a bad earth which causes all sorts of starting issues.

Havent any of your friends got a voltmeter they can borrow you?