If you want TMR and not have to commission an ASIC: Microblaze looks like a good option. For the quantities needed for experiments it would make much more financial sense too. High energy physics likes to play with technological toys though.
White rabbit as an extension of 1588 is a classic example. If they've negotiated a situation where they can can continue to do that: more power to them.
Microsemi (now part of Microchip) makes rad-hardened FPGAs, more so than Xilinx I think, and these days some of them (PolarFire SoC) include a penta-core 600-666 MHz 64 bit RISC-V processor: 4 Linux capable cores, plus one core without FPU, MMU, caches, and able to disable branch prediction, for real-time control purposes.
