I have a big lavender that I love as well - comes back great every year. Mine also is in a pot, on a deck. But I do have that silver reflective paint and no shade, so the situation is probably similar. I have to water mine pretty heavily when the Sun starts blasting. I've not had any problems otherwise, but like NYgardener suggested, I fertiliZe every so often. (I just use Miracle Grow.) Maybe just a little shot of food now and then will do it.

Rooftops in full sun can be hell on plants, even tough ones like lavender, especially young plants in smaller containers or newly transplanted clumps that don't have a lot of roots.

I'm guessing once a week is not enough water given some of the hot days we've had, unless you had a large, well-established plant in a big container. What probably happened is the roots started drying up and the plant started looking ill, and, with the best of intentions, you watered it more to keep it healthy, but with less roots to soak up the water it stayed wet and the roots rotted.

Lavenders are fairly, cheap, I'd start again rather than nurse it back from a near-death experience.

Good luck!

Jim