I'm sorry, I meant to reply sooner and forgot.

In the top waveform at #39, did you still have the freewheeling diode right across the solenoid coil?
In that circuit, it it is as I think it is, the conventional diode on the breadboard would really only protect against the drain of the FET being pulled negative with respect to ground. It looks a bit like it may have gone into reverse breakdown at around 300 V. Generally diodes not intended to be avalanched don't like it much and are given to failing short-circuit.

It is hard to come up with a power rating for a TVS. Given the repetition rate, it is very likely that the average power rating really isn't important, but the peak current rating is. If the diode across the solenoid coil is used, then it will handle most of the energy stored in the solenoid's inductance, leaving only a small amount of energy for the TVS across the FET. I presume, however, that in the interest of parts count it may be preferable to omit the diode across the coil and let the TVS handle all the energy. Most of the energy will actually be dissipated in the coil's resistance.

The peak current in the TVS will be equal to the peak current in the FET at the instant of turn-off, so about 180/1111 = 162 mA.

Unfortunately, TVS diodes often aren't terribly well specified. There will be numeric values for "standoff voltage" and minimum and maximum breakdown voltages and for peak clamping voltage, but the the matter at hand, the latter is at higher current than required. It is safe to assume the clamping voltage will therefore be lower than spec'd but in the absence of curves, the amount is not easy to predict. The larger the die, the lower the clamping voltage for the same current.

The SMBJ170D might be appropriate. The standoff voltage is nominally only 170 V, but the minimum breakdown is 192 V. The maximum clamping voltage is 270 spec'd at 2.33 A (from Vishay datasheet). The standoff voltage may be too low. The next rating up has a clamping voltage well above the FET;s rating. I wouldn't be too concerned given the peak clamping current requirement and the fact that if a FET manufacturer rates a part at 250 V, it will be safe at a full 250 V - but I'd prefer to see some curves for the TVS.

This, from Littelfuse has some helpful general info. I can't find anything at Vishay that is similar, though I know the company that Vishay acquired that produced the original TranZorbs had some useful ap notes. Bourns may have something useful.
www.littelfuse.com/~/media/electronics/product.../littelfuse_tvs_diode_catalog.pdf.pdf

 

In the matter of filtering, for the application I would definitely use a filter cap to avoid added timing uncertainty.

Also, before abandoning simple freewheeling diodes, I suggest trying some fast recovery diodes (no spec for the 200 V part you've experimented with, but it is probably fast; UF4004 should be OK and available in an SM version). To put them on the PCB instead of at the coils, you'll need to add a sort of "clamping rail" - just the 180 V supply but with a small electrolytic capacitor and at least 100 nF of low-inductance capacitor(s) such as ceramic or non-inductive film (prob. through-hole only for the latter). I'd try to have one within a couple of centimetres of each FET to keep the inductance low. Again, the loop areas are important, so the negative side of the caps should similarly be within a couple of cm of the sources of the FETs.