I don't think this combination is possible.
"Reliable" rules out a Mini convertible or a 500 Abarth cabrio. I also don't consider the latter a convertible.
So I think your best bet is a VW Beetle. VWs now tend to be reliable for 4-6 years. You could even get a new 1.8T for $25,000, though a used 2013-2014 with the 2.0T and sport suspension would be considerably more fun to drive. If you want high mpg, these are also offered with a diesel.
The VW Eos hardtop convertible is another option. It's heavier and less agile than the Beetle, but more comfortable and luxurious. Our limited reliability data suggest these do okay, but the complicated roof seems to have the potential for expensive repairs.
A Ford Mustang is a third possibility. They've been quite reliable based on our survey. The main downside is that the Mustang is much larger and heavier than a Miata, and as a result might not deliver the handling you're looking for.
For other alternatives, such as the Infiniti G37, you'd have to either go older or spend much more money.
So, what do you think of these three?