An FPGA is the right hardware. But you won't be able to use the synchronous-focused synthesis software, because it makes the wrong transformations.
For example, an FPGA is perfectly capable of forming an oscillator built with an inverter chain. But if you define that inverter chain in e.g. VHDL and use one of the standard compilers, "NOT-gate pushback" will eliminate inverters pairwise and leave you with only one, and the device won't oscillate.
You may have to write some of your own synthesis software, which will be possible if you get enough information on the bitstream. I'd look into other research efforts that operate on the bitstream rather than the behavioral description -- things like glitch detection and reliability analyses are highly dependent on the mapping chosen by the synthesizer. Probably some work in the area of redundant fault-tolerant logic has already worked out some custom mapping techniques, since common product term elimination is one of the standard transformations performed by a traditional synthesizer, and absolutely destroys a redundant design.
When you control the usage of the FPGA logic element primitives such as lookup tables and local and global interconnect, you'll be able to use the inherent delays to realize your asynchronous design. Your optimization problem is a lot more difficult than fitting with the goal of meeting setup-and-hold times, but that's what makes it research.