All lenses are "SR compatible" … it's the Pentax body that has the SR mechanism
For "extreme budget" telephoto there's a lot to be said for the older FA 100-300mm f/4.5-5.6 and it's variants … mine cost me £2, but it really needs a lens hood, which cost more
Third party companies like Sigma and Tamron should also be considered … often cheaper than their Pentax "equivalents" … again, try to get one with a lens hood if it's a custom bayonet fit.
Even one of the early Sigma 170-500mm "bigmas" might be found within budget if you're lucky