I don't know what Central Time is. I use Astrodient which picks the time zone on its own. So, I used 5:10 AM your local time. I hope this is correct.
ASC is at 14:43 Capricorn, which makes you Saturn Rx at 6:51 Leo in the 7th house. This is descriptive of the situation since it shows your concern about your current relationship. Saturn is in rx motion and in its sign of detriment showing you in a weak position.
Venus is on the ASC also showing concern with love and relationships. Venus is also the significator of your daughter in this chart (5th house in Taurus) which also shows your concern about her. The fact that Venus signifies both sides of your problem shows that both your your relationship with your boyfriend and that with your daughter are equally important to you.
Your boyfriend's significator is the Moon in Cancer 9:32 in the 6th house (and his turned 12th house). The Moon is in a cadent house either way so it shows some trouble, but dignified by rulership, which makes his significator stronger than yours. The Moon is also a co-significator or the situation as a whole.
Like I've already said, tour daughter is signified by Venus. Venus is in the 1st house in the chart showing some dependency on you. It's also in the turned 9th house, but I don't know how to interpret that. Mars in technically in the 5th house that rules your daughter. Maybe she's having some trouble in her life?
The next aspect of the Moon is an opposition with Venus. I think you may be about to have some disagreements with both your daughter and your boyfriend. However, the last aspect the Moon make is a sextile with Mars.
Mars is the natural significator of fights and trouble. In this chart it also rules your 3rd and 11th houses. They are your boyfriend's turned 4th and 9th and your daughter's turned 6th and 11th. It's also technically in your daughter's 1st house.
Here's what I think this chart is saying. You want to please both your boyfriend and your daughter. But because you are having trouble resolving this issue, you may go through some hot arguments with both of them in the process of trying to work this out. However, communication is the only way to handle it. Maybe a friend of yours will help with that. You need to make it clear to your boyfriend that your daughter should feel welcome at her mom's, and you need to have your daughter depend less on you. She should get more friends.
So, although there is not direct aspect between your significator (Saturn) and his (Moon), I guess the answer to your question is stay but there will be some trouble until your resolve this issue.
Away from the chart, I think a good solution would be that you work out some sort of routine where you can have your daughter over once or twice a week, maybe for dinner or something. On those days, your boyfriend cannot complain. But the rest of the week, maybe you and her should talk on the phone or simply go out. Afterall, the man does have the right to have some privacy.
Hope this helps.
It's important to distinguish between means and ends.