I think you're pretty much stuck with copper for your device due to the thermal load you're going to put on it, though this is just a guess. Your 2 minute+ rest between pulses may spare you that necessity, but I don't have a good feel for the thermal issues unfortunately. The key calculation to do is figure out if the RF and electron bombardment heat load melts anything or creates so much thermal stress that something breaks during your pulse.
Frankly, I don't think your financial resource is adequate for designing a new tube, not at all, because after you designed it you'd then have to build it, and I think that'd cost you $30k minimum just to build the tube, forget about the power supplies, cathodes, etc. I think your best bet is to salvage a tube from a decomissioning accelerator and use that.
Since your application smacks of communications, are you sure you can make do with an oscillator? Or were you planning to pulse it on and off like Morse code?
Also, I do NOT think you can count on fixing a tube when you melt it. These tubes operate under vacuum and tearing it down and putting in new parts is going to be both slow and expensive, and you don't seem like you have the cash.
We haven't previously spoken of the Russians, which is a mistake on my part. The Russians like to say they've done everything in vacuum tubes before the rest of the world thinks of it, and they're not entirely lying. They might well have a used tube sitting around somewhere that might SORT OF meet your requirement.
I think in the end though, no matter what you do, you're going to have to compromise on your requirements a bit to get something affordable on your budget, either that or 10x your budget.
There was some RF stuff being done in Brazil a long time ago, i think it was academic work, maybe you could talk to them and get a cheap piece of used equipment.....
As for me, I guess "physicist" is pretty close.
Good luck with your project!