As @Neil_UK cautions, be mindful of the SOA. Lack of a DC curve is not necessarily indication of failure in that region, it just means they didn't test and document it.
At least historically speaking, HEXFETs like the IRFxxx series have been used regularly for linear applications, including RF amplifiers up into the shortwave band. I don't have a problem with this component choice, but due diligence would be encouraged, especially in production.
Testing would consist of taking a sampling of parts, and operating them at maximum required Vds, Id for long enough to thermally stabilize; with a heatsink, perhaps a few minutes. Do this sampling and testing periodically, and especially following any substitution by different manufacturer, product change notification (PCN), or fallout from production test or user failures.
If production is N/A, testing is also N/A. It's a one (or few)-off, who cares, it worked once that's enough.
Anyway, regarding the circuit itself: consider using a Vbe multiplier motif, with bypass capacitors. This will allow the MOSFETs to saturate nearly to the supply rails, and the multiplier can be adjusted by trimmer to set desired bias current. Note that Vgs(th) varies substantially between individual parts. You will not be able to set bias current with a fixed diode stack. Thermally couple the transistor to the MOSFET heatsink, so that its Vbe tracks with Vgs(th) (which similarly has a negative tempco). Doubtful they track correctly, anyway, so do testing at elevated temperature to check that bias current remains stable, and if it's rising too much, consider introducing an NTC to the bias network as well. It will take some back-and-forth testing to figure out the best values.
Beware of oscillation. A capacitive load can be quite hazardous to many amplifier designs. Without global feedback (i.e., the op-amps don't take feedback from the final output, but merely from their own outputs), there at least shouldn't be oscillation due to op-amp compensation (or the lack thereof); the transistors can still oscillate, in and of themselves, particularly due to poor layout and the combination of load and device capacitances making some manner of Colpitts oscillator. (Use ground-plane layout techniques, and bypass the supply rails with low ESR capacitors. Actually, or maybe don't; relatively high ESR may help, or ferrite beads on the drains. It depends.)
Also, do you know you need bipolar supplies and bridged mode operation? It seems a bit redundant. Higher supply voltage may be feasible
Exceptions could be: perhaps the op-amp can't handle enough voltage, or suitable amps can't supply enough current, or are too slow or too expensive, or doesn't even exist and a discrete solution would be required. I haven't looked into parts of this capability in a while so I don't know offhand what might (or might not) be suitable.
Another option might be using a single output (half bridge) with a relatively low-Q matching network, to increase the load voltage modestly, while maintaining desired bandwidth.
Also, do you know that you'll achieve desired bandwidth in the first place? Typically, piezo devices have numerous resonances at high frequencies, and it may not be clear that you'll achieve the designed mechanical bandwidth. But it could be the "400kHz-1MHz" spec is merely electrical, and mechanical is just whatever, I don't know. Just putting this out there.