I think for this one night yes, invite him. Just because it's a special birthday more than anything.... 18 and all that.
I know it won't be easy but it can be done. My birthday is boxing day and for years after my mum and dads split, dad would stay over on christmas day in order to spend my birthday with me and I don't suppose it was easy for either of my parents. They're both now re-married and now we quite often go for meals with their respective partners and my sisters on special occasions. It does get easier with time and probably as your kids get older.
I don't think it's something that can happen all time and that needs to be explained to your girls as you've obviously got a life too which includes your current fella and maybe you need to explore with them why they don't like him so much, (possibly they're just not used to him or they miss the ex and that's all pretty natural and again, gets easier with time).
And make sure you've got a day set aside for you and your partner to get over the whole messy event and spend some quality time together. It's your life too. Just a special occasion and almost a special treat for your daughter for this one night.