have you thought of flying into Bari (BRI) which has a direct route from Madrid (and also from LGW with BA.)

You can then catch the ferry direct across to dubrovnik, its about 7 hours i think - but could be quite a relaxing way to start a holiday

this is the main company:
http://www.jadrolinija.hr/default.aspx?lang=2

but i think there are a couple of others.

i don't think you will have much jioy with low cost arilines to that area out of heathrow, you'll end up going to luton / stansted. LHR to LGW does not have to be a nightmare. just make sure you leave plenty of time and don't do it 8am or 5pm. the coach will drop you terminal to terminal pretty much!