First, given the impedance of the K7XX you’re going to want to stick to either tube-hybrid amps or a tube amp that has an output transformer.
The K7XX cannot be driven balanced so you’ll also want to stick to a single-ended amplifier (otherwise you’re either paying more than you need to for circuitry you won’t be using or you’ll be getting lower-quality parts than an equivalent single-ended design could manage for the same price).
The only competent all-in-one DAC/amp that uses tubes (it’s a tube-hybrid) and that comes in around $300 (it’s $320) that I know of is the Massdrop/Drop x Cavalli Tube Hybrid w/ built-in Grace SDAC. There may be others, but I can’t think of any off the top of my head.
Otherwise …
To do a DAC and a tube-amp for a total $300 is going to be a bit of a challenge.
The DAC part is the easiest to tackle … either a Schiit Modi 3 (which would be my choice to pair with the K7XX) or the Massdrop/Drop x Grace SDAC.
The amp part is harder … I cannot name a single pure-tube amplifier that I would recommend for $200 (doesn’t mean there aren’t any, just that I’ve not crossed paths with one). That means tube-hybrids - all of which will have less of a “tube sound” than pure tube-amps.
But, for tube-hybrids within your budget, that actually have a reasonable element of “tube sound”, I am partial to the Schiit Vali 2. The others that I’m aware of will break your budget unless purchased used.
It’s a bit hard to follow your requirements … so one thing I note is that after you talk about having a dedicated all-in-one for your headphones, it also sounds like you want to use it as a source/pre-amp for your speakers. The Vali 2 will fulfill that criteria, but the Cavalli Tube Hybrid I mentioned above does not have pre-outs, so it wouldn’t.
You also talk about the DAC having “higher capabilities” down the road. Pretty much all modern DACs can support high-resolution files, including the two I suggested above (of which the Schiit will go to 24/192, the Grace to 24/96). If you want DSD support you’ll need to look at other products. If you want MQA support, you’ll need to up your budget or compromise on the overall quality of the DAC.
Finally, I wouldn’t worry about having a separate DAC/amp and using RCA cables to connect them. You’re only going to need something maybe 6" to a 1’ long, and you’re not going to be able to tell the difference between that and some cheap internal wire with this stuff anyway.