I am sure you thought of it but it is not doable - the easiest way would be to thru-check the bag and just walk on the air side if you are arriving domestic. All the terminals at LAX are connected now although the up and down/tunnels can be confusing. If it is international -> domestic and the bags are tagged through, just drop it off after customs.
I have made pretty short connections from TBIT to UA but it involved a lot of running. To me, 4 pieces at 60 lbs each is a lot to push yourself but taking a taxi can makes no sense either - might take time to get one and they probably will not like the very short ride. So, it seems the best way is to get a willing skycap to do it.
Time of the day might also make a difference - the curb can get really crowded, especially at TBIT during certain departure banks.
My only suggestion is to walk upstairs since the air will be better. I think all the constructions have finished but not 100% sure.
Do you have any status / TSA Precheck on the UA side?