Sounds good. I'm curious, how big is your coil? What's it like?
You wouldn't want only two of the 26-volt TVS because a 55V flyback would be slower than you're looking for, but you could use a modest number of higher-voltage TVS parts. It's also be nice to drop back to 1.5kW parts, which are more commonly available. For example, you could use six of them in series, for a nice 40% safety margin.
Well, if you were to use 450-volts worth of TVS in series, then you'd get a discharge time I*L/V = 2.8ms, likely faster than you need. If you pick a lower flyback voltage you'll have an easier time finding low Ron switches, which is important to avoid a big fan-cooled heatsink. :-) For example, a 175V flyback would give you a 7.3ms shutoff but allow the use of 200V FETs.
Fairchild's FQA34N20 or ST's STW34NB20 (both in stock at Mouser for about $3.20 each), have Rds(on) of about 0.06 ohms, so they'd dissipate only about 4W, which is a big improvement over what you were facing with 500V parts. I'd even consider using one of these (wired backwards) to replace D1, because their 0.5V drop at 8A is better than you can do with a 250V diode. I'd use good heatsinks on the FETs, you want to keep the junction temperature down to avoid an increase in Rds(on).
To get the 175V flyback, you could use six series 1.5ke24a (also called 1p5ke24a, to eliminate the decimal) TVS, which drop 27-30V at 8A. 24c at Mouser. Six of these are rated to absorb 9J in a 1ms pulse, even more in your 7ms pulse, fine for your 5J energy.
As for C1, it's be wise to limit the flyback risetime to say 0.2ms, limiting dV/dt to a modest 0.9V/us. That means a capacitance of C = I*t/V = 9uF. An ordinary 10uF 250V electrolytic would probably work fine, given your presumably-long interval between events.
The current ramp has a V/L slope as it starts, but the coil's high series resistance makes it tail off in the classic manner you're use to seeing with a charging RC. The time constant is L/R, or 0.16/3 = 53ms. But it'll be within 10% of 8A after about 100ms.
Cheeky blighter that I am, I'll accept for both of us. Have fun, and please do report back.