I don't know if that many SET amps will fit your profile. While SET amps are, generally speaking, simple in design, they require really good, and expensive transformers (generally large, air-gapped transformers). The tube type that fits the criteria you mentioned is pretty much limited to the 300B. But, most of the 300B amps are on the warmer sounding side.
The larger transmitter tubes (e.g., 845, 805, 211) run at quite high voltages which would make me very leery about any cheaply built amps. Parallel output tubes in SET configuration is sometimes used to get a bit more power out of the lower-powered triodes, but, that turns out to be not that easy to implement.
I think you will have the best luck looking at pushpull amps. My own personal preference is amps running 6L6 tubes. These put out less wattage than EL34s, 6550, KT-88, etc., but, to me, they sound sweet without being muddy or sluggish sounding. A lot of lower-priced tube amps run EL84s; I like this tube too. I generally like the amps offered by Synthesis (Italian) and Audio Note (UK).
There are a lot of very old vintage amps that can be found in your price range, but, you would have to work with someone who knows, and can refurbish the old stuff. A refurbished RCA or Northern Electric 6L6 amp will blow away surprisingly expensive current offerings. Other old brands, like Scott or Brooks, have really nice models too.