You and your DP need to put your heads together and think of a way to find a more appropriate time to get your 10 minutes together. I find it very hard to believe, make that impossible to believe, that the moment your DP arrives home is the ONLY time you can get your 10 minutes together.
Don't get me wrong - you are doing an awesome thing in teaching out for help. And you are doing very well in taking in what hubbers are saying. Just than for me, these two items together suggest that less focus needs to be on you and your DP's budding relationship and more focus needs to be placed on integrating your DP into your daughters life. Kids love stability and security.