Romances, unfortunately, seem to be core part of companion experience - so your very best chance, is to either turn down the companions when they start flirting with you, or have negative reputations and hear "yo, we could have boned if you did stuff I like. Too bad for you". Hopefully, after the horny Act1 and romance will develop, or be unexistand if you didn't hook up with anyone.
As far as invasiveness, I didn't find it too bad.
