What is CIO?
Several issues here, one of which is that you allude to your DD's weight/height. Is she tiny? Was she prem? If yes - disregard what I say below and speak to your Health Visitor/GP/paediatrician etc.
If she's not, and she's a full-term baby who is not small for her age, then your GP is correct and she shouldn't need feeding at night at 6 months old, as long as she is getting enough milk during the day. Why did he suggest mixing formula and breastmilk? - is it because your DD is not getting enough milk from breastmilk alone?
A baby who is having 5-6 full feeds during the day definitely does not need a feed at night (by which I mean between the hours of 11pm and 7am-ish)!! It's a habit/comfort thing. For both of you. I'd be willing to bet that, if you were formula feeding her at night, you wouldn't be putting up with the extra feeding as you'd have to get up, get the bottle, sit propped up in bed, hold it, etc etc - much more of a palaver than popping her on the boob and nodding off!! (This is not a criticism of you, or of breastfeeding in general, it's just an observation).
It's up to you what you do at this point. However, of course your DP should have a say in it too; she is his daughter too. You can carry on as you are, but your DD won't start sleeping through the night unless you help her. Or you can start to get her into some sort of routine. Gina Ford (can I say that on here?), Baby Whisperer, etc etc. There are plenty of books etc out there that are very helpful.
When are you planning to start your DD on solids? Because she certainly shouldn't be needing a night feed by then. And if she's still only on milk, it does sound as though she's not getting enough during the day. Hence the GP's comment about formula, I'd imagine.
How many feeds a day, and how much expressed milk, is she having? And is she draining your boobs at night too?