I don't see what's wrong with a Brayton cycle, using the airstream as a working fluid...
Doesn't that lead you right back to the problem of how to heat the airstream?
That's what I've been trying to get across to you. Instead of using a REB to heat the air directly, you use it to heat a fluid in a closed loop. This fluid is pumped through a counterflow heat exchanger in the engine, thus heating the air.
The closed fluid loop doesn't need to be a power cycle, so it doesn't have any inherent inefficiency. And as a bonus, on the cold side (after the engine heat exchanger and before the REB) you can put another heat exchanger, to add heat to this loop from another
loop - a heat pump, with its own cold side being the cooling loop from the reactor. (Or you could just have it be
the cooling loop from the reactor, if you can find a suitable fluid...) This allows waste heat to be pumped to the engines along with energy from the REB, solving the problem of what to do with it and increasing the total efficiency of the system.
The heat exchanger would need to be made of something that could take the high temperatures involved without structural compromise. It would most likely be very heavy, but the reactor is incredibly heavy anyway so this might not matter much. And the turbine in a modern jet engine has to take similar temperatures (for obvious reasons), so it's not like we've never done anything like this before...
This scheme won't work at hypersonic velocities, because the heat exchanger would need to get too hot. I suppose you could get fancy with nuclear-lightbulb-style radiative exchange, but the fact is that at the altitude you'd be hypersonic at, ozone isn't a big problem, so you can use the REB directly.
Of course, if you do that, you need another way to get rid of waste heat... all I can come up with right now is start dumping hot hydrogen into the combustion chamber upstream of the REB, like a standard airbreathing ARC-QED drive...
This scheme should give you a vehicle that can fly for an unlimited time at low Mach numbers, but needs to pack LH2 for hypersonic flight. Oh well...
Plus you don't have to deal with multiple GW of electricity, which is difficult even at low voltage.
Looking at large helicopters (CH-53K), the max takeoff weight power-to-weight ratio is somewhere in the neighborhood of 200 W/lb. A heavy 500,000 lb vehicle (widebody airliner class) would need (very roughly) 100MW of lift fan power to leave the ground (ignoring among other things thrust efficiency differences between a single large rotor and multiple smaller fans). I'm assuming the Polywell fusion rate can be modulated (maybe a bad assumption). So I'm not thinking of multiple GW for low altitude operation. If you MUST run a Polywell at full blast, always, then I would use part of the diffused REB to run the Stirlings and dump the rest of the REB into the air (upwards, to avoid lawsuits).
500,000 lb? A fully shielded 6 GW BFR weighs twice that on its own.
A modern ultra-high-bypass turbofan can get over 50,000 lbs of thrust from less than 90 MW. 6 GW gives you 3.4 million lbs of thrust, or about one STS RSRB. That's enough to lift an aircraft weighing about three times what the reactor does. But of course a horizontal-takeoff aircraft doesn't need a 1:1 thrust-to-weight ratio to take off, so perhaps it could be larger. And/or the engines could be smaller, so as to enable supersonic flight.
Or you could just let the reactor be more than a third of the all-up vehicle weight, and get something that flies like a giant Su-37... probably easier said than done...
For military applications, shadow shielding might be worth the increased performance. For space launch, it probably is.
The reason I've been using such high power is that the shielding mass goes up so slowly with output power. Remember, Polywell gross power goes as the seventh power of the linear dimension. Shielding thickness is fairly insensitive to power output (it has to reduce radiation by so many orders of magnitude anyway that a factor of 60 is nothing), so to a first approximation shield mass scales as the second power of the linear dimension. Thus larger reactors have much better power-to-weight ratios, not just for basic structure, but also for shielding.
Assuming the superconducting magnets can't be modulated, how deeply you can throttle a BFR depends on how the losses behave when you detune from nominal operating conditions. Increasing the hydrogen-to-boron mixture ratio would reduce fusion power and bremsstrahlung, the former probably more than the latter. Electron loss rates shouldn't change much. Altering the magrid potential changes the temperature and thus the nominal density; it also changes the fusion cross section and should affect bremsstrahlung slightly. I think you should be able to get it fairly far down, maybe as low as 20% output including waste heat, before you start having trouble running ancillary systems like the cooling refrigeration loop, but of course this is essentially a guess.