Yes, I do believe that one week is not enough time for a poor diet to effect the fish. That being said, it probably wasn't being fed a proper diet at the LFS where it was bought either, especially if the OP was not made aware of its care requirements. I've seen many LFS's that do not feed anything more than brine shrimp to their fish in hopes that they'll be bought before they're completely malnourished.
To the OP, if you decide to go with another Flame, or any other dwarf, I'd wait a few more months until you're tank is more stable and mature. In the mean time, get good test kits and learn how to use them. Dwarf angels, as well as many other fish are very sensitive to poor water quality. Also, I do no suggest another dwarf for this size tank, it's going to stress them out. If there was something wrong with your fish ahead of time, the small tank might have stressed them out just enough to make a difference. Unfortunately, I do not think you'll ever get a definitive answer on the cause of death, but here's what I can say to you:
Poor Water Quality
Small Tank
Improper Diet
Under established tank