With 135mm lenses it is hard to go wrong. Even the lowly 3-layer coated Takumar bayonet 135mm f2.5 or f2.8 are OK or even good. Or any of the Takumar 135mms. Also many 3rd party 135's are OK. It is a simple range to make a good lens, or if it is so that they had a lot of time and practise to design them. Since everyone had one in their bag back in the film era many were made and they can be had cheaply.
There is only one autofocus Pentax 135mm, and you wont get that one for us$100. But there are two alternatives that can be had with A contacts: the SMC Pentax-A 135mm f2.8 and the Tamron adaptall 2 135mm f2.5. Both are good lenses. The Tamron even a bit better. Note that if the Tamron comes with a K mount adapter or some other brand mount you can switch to a KA adapter.
The 200mms were also made in fairly large numbers, but probably only 1/10th as many as the 135mm lenses. Among the Pentax offerings you will have to stay with manual focus and have more patience to find one in your budget. Pentax changed optical design between the K200/4 and M200/4, but both are good enough.
With tele zooms it is more of a gamble. Cheap often means old and early zooms were often not that good. Much more have happened in the last 20-50 years when it comes to zoom optics than it has for primes. Among the Pentax offerings in the range 70 or 80mm to 200 or 210mm the best options (not counting the FA*80-200/2.8) is the M80-200/4.5 and A70-210/4, both constant apperture, and the later even with A contacts. The F 70-210 you mentioned is just OK, but not as good as the M and A alternatives. Stay away from the F 80-200 and FA 70/80-200/210's, including the powerzoom version.
If you go 3rd party and manual focus, consider the Vivitar series 1. They are good for their age. There are 4 or 5 optical deisgns. Several of them with a macro feature and the last one with A contacts. My favo is the original 70-210 f3.5.
Same applies to most 70/80-300mm lenses and the range 200-300mm is typically worse than the rest, less sharp, less contrast. Knowing that, Tamrons various 70-300 are probably as much as you will get for the bucks. The DA55-300mm is a better lens that stays good even up to about 270mm. But it is out of your budget.
Why not get a cheap 135mm for half your budget now...and then save up for a used DA55-300mm?
And with patience and luck you can solve almost any lense need. I got a Sigma 400 f5.6 with A contacts and autofocus for -60 us$ on ebay. Was sold as "not working"...but I saw on the pics the A contacts were dirty...and decided to take a chance. I don't use that focal length much, but thought I it would be fun to have one for wild animals etc. Won it, it arrived...I cleaned the contacts and it worked.