From personal experience
1. 500PF
2. 300PF
3. 300PF + 1.4 TC
4. 200-500
They are all good lenses just that some are "gooder" than others.
Perfectly agree on this one, but ...
My guess is you won’t find a used 500PF within your budget.
Did you consider a 500 F/4?
Sharp, REALLY usefull with a 1.4TC.
A 500PF is reported to be somewhat usefull at F/8, but I wouldn’t recommend to use that combo regularly.
The 500 F/4 however is used by a lot of shooters with the TC ‘glued’ to it. (700mm @5.6)
I have both the 500 f5.6 PF and 500 f4 G VR and I love them for different reasons.
The 500PF - and probably anybody having used one can confirm - is a breeze to shoot handheld and IQ is great. If mobility and agility is important for you this might be the way to go, especially if 500mm is enough and you don't have to bother about using a TC. With a TC you go to f8 and thus loose not only AF speed but also most of the AF sensors which can be a challenge with BIF shooting. Oh, and I ignored a potential issue with the budget, because there is a reason why these things are difficult to get ...
Not knowing about the market in your country it would be still difficult to get a 500 f4 G VR here in Germany for $ 2500. They are more in a range around € 4000 = around 4500.
If you find a good one for a reasonable price you get a great lens that alsos lives in perfect harmony with the TC-14EII (use it myself) and TC-14EIII (have heard of), which would give you a 700mm f5.6 with great IQ, AF speed and accuracy. The main disadvantage here is the weight and front-heaviness of this lens, which makes it difficult to handhold for a longer period. We are talking about 3,9 kg here plus body. If you get a good one for a reasonable price you should look for getting a support system, i.e. a tripod or a monopod (see
@Steve 's recommendations to choose). Upon his recommendations I just got a monopod with a Wimbrley MH-100 for the 500 f4 and it's great to use and it is cheaper than an adequat tripod with a Gimbal.
One thing I certainly wouldn't do these days even if I have to stretch my budget a bit and this is buying a tele prime without VR.
Oh BTW: I had the 200-500 here side by side with the 500PF and I sent back the 200-500 after three days. Reasons were that I found myself shooting at the long end most of the time anyway, that the handling of the 200-500 didn't suit me - but that's personal preference only - and I wasn't really happy with the AF speed when comparing it with my other lenses.
But there are a lot of people here that are happy with this lens and it fits in you buget even new ...