Since DL will not through check to a non-partner on separate tickets, you will need to exit the secured area and collect your bags at LAX and move them yourself from T2/3 to T7. One of the positive things is DL has a 20 minute checked baggage guarantee, so that would be helpful. As others had pointed out, taxi and Uber are not realistic option, since you'll need to pick up your ride from LAX-it area, which is counter-intuitive.

A few questions:

1) Can you push ~240 lbs by yourself? If not, consider taking the inter-terminal bus, but you'll need to manage 4 overweight bags + 1 child.
2) Is your UA flight domestic or international. That'll determine the baggage cut-off (45 vs 60 minutes)
3) Do you have status on UA.*A, or booked in premium cabin? If not, you'll need to pay checked overweight baggage charge (which I'm guess you will not based on your Premier Access comment)
4) Premier Access desks are closer to T6, so that's to your benefit and will shave off a few minutes.