+ 1 for prev answers. Your "reach" options, of which there are more in this digital age, are
- more focal length - lens and/or teleconverter.
- More pixels.
- smaller sensor (= sensor pixels concentrated in a smaller area).
Lens options at budget indicated are stricly limited. Something like the takumar 500mm f4.5 might cross your path. I can recommend the sigma 500mm f7.2 apo, MF PKA mount, ticklish focus, good iq, it's a bit of a sleeper in the s/h market, can definitely be acquired for less than $200 and doesn't usually suffer from the haze of an internal cemented group that almost always afflicts the contemporary sigma 400mm f5.6 apo. Mirrors already mentioned. Most of the vintage 400's I have tried don't really do better than the 55-300mm. A 1.4 x tc is certainly workable with the 55-300mm but do you really gain anything in the actual results?
More pixels is a good option but a more recent pentax dslr is unlkely to be had for $200.
Smaller sensor - a Q is well worth considering, married to a high res lens like a 90mm macro gives you pushing 500mm equivalent. And +1 for a superzoom/bridge camera with eg 600mm/800mm equiv as a realistic and very practical, easy lightweight option. Photographically there are slimitations, they won't normally do RAW and the internal processing can be not what you might want (biased towards contrasty postcard sort of look IME...). And these options give enormous depth of field compared to long lenses on apsc/full frame, big advantage with eg pics of flocks of birds.
And then there's digiscoping. Previously my efforts with a HR66 scope by opticron were disappointing, but I recently got a HR80 - an old model - for around your budget and this is better. TBH to get best results with digiscoping then you really need the best scopes eg Swarowski, Leica, so really much depends on what you want to achieve.
Pics: HR80 + K3, opticron 41111 digi adapter (eyepeice projection), 27x HDF eyepiece. RAW, cropped and pp, 1:1 pixel peep. Worth bearing in mind the effective f stop is something like f32, well past normal diffraction limits.
The digiscoping mode I haven't really looked into (mainly bc am one of the diminishing band of luddites w/o a smart phone) is photographing the exit pupil from the scope with a small sensor camera. I did a bit on my HR66 with a panasonic LX5, (has a good optic) but wasn't impressed.