We did old Route 66 west-to-east this past June, and the Cabriolet handled the desert running from Barstow east just fine.... Had a ball climbing the hills around Oatman AZ through the Sitgreaves Pass... No trouble at all crossing the Mojave and climbing whatever had to be climbed, with the A/C on in the 100+-degree heat and the cruise set to 75+... CA, AZ, and NM went by as easy as you please.
I will say I'm glad I had put sound deadener and some extra insulation under the carpeting, especially over the engine compartment - the cabrio top doesn't insulate very well in the high desert in high summer, and we did have some noticable heat transfers through the cloth roof between the air-conditioned interior and the hot-as-blue-blazes outside air. But we didn't have a heat issue at all from radiant engine heat coming through the rear compartment carpeting.
I put just under 11,000 miles on the car in ten weeks this spring and early summer, all of it very comfortable. Your little jaunt from Vegas to L.A. should be smooth sailing. You'll want to keep an eye on the winds in the Cajon Pass, and if they post high-wind warnings you might wanna take a rest stop for a bit, but aside from that, no worries.
I've been doing ~40-41mpg this summer, with the cruise at 65 and the A/C on. You won't do as well on a new engine, but as they age and wear in, the fuel economy tends to creep up. A year ago, I would be lucky to see 35mpg under identical conditions.