It's a matter of supply, demand and ability. The k series cpus are far more prevalent on the used market, and quite often can be cheaper than their non-k counterparts as a result.
There's absolutely no reason why a K should be excluded from searches. Just because Op has a non overclocking motherboard does not mean he can't take advantage of the higher stock speeds on the K's. And that doesn't preclude any OC that can be applied via BCLK and native software.
There's only one thing that could prevent Op from any and all worthwhile upgrades and thats the motherboards bios. Dell, Acer, Lenovo, HP, Compaq, Sony and any other 3rd party OEM pre-built will by design have proprietary bios restrictions, most normally preventing the use of any compatible cpu that's not included in the model line. Xeons and K cpus being quite commonly not supported.
Naturally, the i7's will be the better value, being the most capable all around with 4c 8t, followed closely by the i5's. The other pentium and i3 cpus are so close in ability, and so lacking in such things as Lcache, that even with 4t ability they still come up way short of an i5.
Because of Op's budget restrictions, it's going to take some serious wheeling and dealing and a whole Lotta luck to get anything worthwhile, but the deals can be had. I picked up my i7-3770K for US$50, simply because the seller mistyped the add and listed it as a I3-7770K, 5 years ago when Haswell was still king and a used i7 was $200+. Fractal design define R5, brand new, had a small dent in the off side panel (took 3 minutes and a 2x4 to fix, flawless) US$20.
You can find the deals, just gotta be willing to look, and think outside the box.