Reddit mentions of S-Engine USB MIDI Sound Module

This synth-action piano from Alesis fits your budget - with room for a cheap multi-effect pedal on top. (A multiFX is probably the quickest way of turning five relatively undistinguished sounds into something a bit more unique. Within your budget, these kinds of GM boxes are your alternative for expanding your palette in a relatively portable way, perhaps combined with a slightly cheaper dumb keyboard like the Keystation 88 mk2; but chances are you'd still want the multiFX, and that quickly turns into Too Many Boxes...)

edit: If you can stand to lose an octave, Casio do a few 76-key keyboards (the WK245 is $199) - but at least one commenter complains that the black keys are too thin, so maybe try one first if you can. That might be your best option for something you can sling under your arm and vamp on with a few friends.

A good piano software VST will be somewhat demanding, so a Raspberry PI or Intel NUC will probably not provide optimal latency/polyphony without clipping.

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I was in a similar situation -- good action, bad onboard sounds. My solution was to use Pianoteq on an older Macbook Pro (total cost: $600) and I can produce sounds that I like, with roundtrip latency around 3ms. I shopped for a used Macbook with a damaged screen, so that my dollars could go toward my ears instead of my eyes. A hardware MIDI module beats the latency problems, but you'll have to pay quite a bit to get a decent piano sound, and the UI will likely be annoying. They make modules that will technically fulfill your requirements for under $100 (https://www.amazon.com/gp/product/B01E5GGKE4/) but they don't sound great, or you could spend ten or twenty times that, and get a full-featured rack module (https://www.amazon.com/Yamaha-Motif-Mount-Sound-Module/dp/B0019X08CM/).