Throid mé agus bhuaigh mé means “I fought and I won”, and is seems to be perfectly fine sentence.
Throid mé agus chloígh mé would mean “I fought and I cleaved, adhered…” according to
Focloir Gaeilge-Béarla. The word
cloígh can mean ‘to overthrow (an enemy)’, but according to that dictionary only as a transitive verb, so it needs an object, eg.
throid mé agus chloígh mé mo namhaid / an impireacht would mean ‘I fought and I overthrew my enemy / the empire’.
So I believe
throid mé agus bhuaigh mé (or, if you’d like that, more archaic and Munsterish
do throideas agus do bhuas, or shorter, more modern Munster:
throideas agus bhuas) would be better.
But, anyway, I am a beginning learner and I base my opinion on pretty much single dictionary, so I might be wrong, and other people here may know better words and constructions for a better, more idiomatic translation.