Unfortunately, your Ex has the same rights as a full time Dad (which as you said you dont mind) but the fact that he has a girlfriend, there is not much you can do about this. Your daughter's upbringing if your ex wants to be involved, is between you and him, nothing to do with his girlfriend at all, your DD already has a Mum, you & your ex are her parents!!!!!
Don't think for one minute the fact that they are a couple and with him being the Father that they have the upper hand either, it doesn't work like that................
Obviously the fact that he would be a stranger to your DD would mean that you could seek advice and get supervised visitations, then take it from there, but it is very unlikely he could turn up tomorrow to take her for the day. But on a good note, there is no way he would get custody of her as she has been with you and only you for the past 6 months and at this age it is very important for a baby to be with it's mother, a court always favours the Mother except in extreme cases! You would have to be proved unfit, as already said!
But if he states he will not see DD without his GF being involved, he doesn't sound very dedicated.......so personally I would go see a family lawyer (you get an hour free for consultations with many, so look around for them) write down every question you have before hand and get it answered.......if you need further help and they will take on your case, then you can apply for legal aid (if you are entitled) and possibly with one whiff of a legal situation going on, he may lose interest?! If not, at the very least you will have to agree terms and they would be put down on paper and if he doesn't stick to them, you have a stronger case for any future problems and you can stop the visitations as he is not sticking to the agreed visitation terms, so you would be well within your rights to do so.
Please don't think I am advising he has nothing to do with her full stop, but you did say that they are both no good for her........you know these people, so I am just giving advice on possible options for you!
HTH
Good Luck hun xx