You should be fine. It's only a few minutes walk from T1 to T7 (T8 security is through T7). If you don't have luggage, or have only a roller bag it's pretty easy. You could probably easily do it with one of the carts, too, if you have a lot of luggage. You have to cross at the lower level, and on the sidewalks at East Way. It used to be possible to cut through the parking garages at both the upper and lower levels, but the connections at the upper level are cut and the lower level is blocked by construction barriers, so you have to take the real sidewalks now. If you really want a walking tour of the airport and don't have checked bags, you should have enough time to do the whole trip airside, including getting lost once or twice in the basement tunnels.

I don't think I've ever gone though precheck at T7 that late, so I don't know how busy it will be, but I've never really had much of a line for precheck in T7 at other times.

I routinely park in P1 (the parking for T1) and walk to T7 and T6 because it's easier traffic in/out. There's also a bus that goes around the bus loop between the curb and the islands, but if your not mobility impaired it's generally faster to walk from T1 to T7 because it stops at every terminal. When there are a lot of people on the Flyaway bus or the parking bus, I'll get off at T1 and walk across to T7 to save time.

ETA: there's even an interactive map that will guide you the whole route airside, if you want to try it: https://www.flylax.com/lax%20new-terminal-maps