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You have made several serious errors that invalidate your point.
a) The facts that you need high temperature to achieve high exhaust velocity, and that there are limits to the temperature of solids, do not imply that high exhaust velocity is impossible. All you need to do is ensure that your rocket motor is not in thermal equilibrium with your propellant or your fuel. The obvious way to do this is to have low thrust, as that allows your cooling system to keep up. Making your fuel also your propellant, and limiting its ability to thermalise before leaving the ship, also help (the limit is still proportional to F*Ve, but you can raise the proportionality constant). It helps a lot that plasma can be contained magnetically.
b) Thrust doesn't matter all that much except for takeoff. The time taken on a brachistochrone trajectory goes roughly as the inverse square root of your thrust (1), so a millionth the thrust is only a thousand times the travel time. 1 mG could get you the distance to Pluto in about a year and a half. What really matters for transit time is the delta-V needed to achieve brachistochrone at all, and that means nuclear.
c) Speaking of which, yes, nuclear has staggeringly-higher Isp than chemical, to the point that the rocket equation is generally in the linear rather than exponential regime for interplanetary flight and, hence, its "tyranny" is indeed "escaped" (interstellar's a different beast; if you want to go relativistic you're generally looking at antimatter fuel or external drives like light sails). "800 seconds" is a fucking joke compared to what nuclear's capable of - in the highest-Isp version of fission, the fission-fragment rocket, it's capable of millions.
I made the point about exhaust velocities mostly wrt fission engines using thermal gas as a propellant. If you have a fission reactor as an energy source, getting a heating your propellant to a much higher temperature than your fuel elements seems challenging.
The length of your brachistochrone would depend on your available acceleration. If you have unlimited thrust, the fastest path is a straight line (relativity aside). If your thrust is very limited, I would expect that you will spend a lot of time orbiting the Earth while prograding until you escape it eventually, and then you will spend a lot of time circling the sun until your intercept.
However, I agree that 10mm/s^2 is still a usable amount of thrust within the solar system. The Dawn spacecraft got around with much weaker engines, but it definitely increased the transition time.
I am more skeptical about fission fragment engines. Sure, the exhaust velocity -- a few percent of c -- is amazing. But for every fragment which escapes and generates thrust, another one (or three) will hit your spacecraft. Because energy scales with v^2, if you want a decent thrust, that will mean an ungodly amount of energy. 1kN times 0.01c is something like a few Gigawatts of thermal power, similar to what a large commercial nuclear reactor might have. Cooling this away in space would be challenging. And if you add a reaction gas to get more momentum per energy (at lower exhaust velocities), you still have to confine that gas magnetically, which also adds overhead.
Right. So you run open-cycle. You mix your fuel with your propellant, stick it into your nozzle, and burn it at plasma temperatures (remember, while terrestrial nuclear reactors burn up their fuel over the course of years, this is not actually required; a nuclear bomb burns its fuel to reasonable completion inside a microsecond). You will need cooling systems for the nozzle, of course, but the temperature gradients are all in the right direction.
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