Thanks for the replies so far. From the Eva Air website, it says
"In addition to the hand-carry baggage (7 kg limit), pax may carry, without additional charge, the following carry-on items:
1. One pc of laptop or purse or small backpack or briefcase or camera case
2. Reasonable quantity of coat, hat, scarf, shopping bag, food, drink, and other personal belongings."

Given that, I suppose I can put some items (like a tuxedo, which is heavy) into a shopping bag and bring that in along with the carry on bag plus the computer/personal bag. Once pass security, I can then repack the carry on bag with the tuxedo from the shopping bag.

The reason I don't wish to check in items is the risk of lost or delayed baggage (which would be very difficult for me on this trip), or having to wait an inordinate amount of time at the baggage carousel at LAX.

I'll see how it goes; at the worst, if I have to check the carry on bag, then I will check that in rather than getting into a big battle with the check in staff.