After only a few tries I found a strategy that seems to result in my dominating victory (on normal skill level) but though I have space supremacy, invading the enemy if he's got a fully-developed planet is very costly. Apologies - the numbers below come from memory so may be quite rough.
1) I have fleet size 15 (quite advanced logistics) but this results in a maximum invasion force of 3000 (three cargo ships each with two troop modules). It takes at least two of these fleets one after the other to conquer a planet of 23bn even with space superiority and superior weapons.
My enemy has four or five planets of this size - making enough ships to carry the troops to invade on this scale is hugely expensive even when I have by far the best economy in the galaxy.
2) Do the soldiers have to butcher the planet's inhabitants in order to win? I easily conquered a minor race on a planet with 18bn inhabitants but when I was done there were only .5bn left! Could there be a 'tactic' which minimised civilian casualties by only killing soldiers and leaving a subject population behind?