I worked in a DPI/firewall company and my work ran on the ASIC accelerator, so nah, my 'guess' is probably better.
FPGA is not worth the trouble. You get neither the (line) speed of ASIC, nor the flexibility of running everything in the CPU. Most serious DPI hardware vendors have stopped using it.
But you are right that it's no fun trying to workaround ASIC bugs.
FGPAs allow near-ASIC speeds with effectively the flexibility of software in that they can be updated via firmware upgrades, with much cheaper dev. costs than ASICs. They do have a higher unit cost than ASICs but only at high volume. For anything that is 'low' volume an ASIC may not make financial sense at all in any case.
I am no expert in DPI specifically but Google suggests that using FPGAs for DPI is an active commercial topic.