Gertrude, I think there may be two categories of US airports for transit purposes. There are those in the SE for which a significant minority of passengers are in transit between Europe and other parts of the Americas - and are therefore tolerably well-equipped to handle them.
Then there are the others for whom transit passengers are only a tiny minority and that therefore don't deal with them quite so gracefully. I would place LAX in the second category and, reading between the lines (30hr flight, SIN as alternative transit point) I guess that's were you'll be.
So be prepared for some inefficiencies, but if it were me, I'd pocket the £300 saving and just grin and bear it. (What else are you going to do with your transit time, anyway?)
SIN is at the other end of the scale, with a very large minority if not an actual majority of PAX in transit (true for some ME airports as well) and I can't think of a more pleasant or efficient place for intercontinental transit. The big plus over the States is that you aren't dealt with as a potential immigrant!