Are you using one? Experience?
Not a 2000 series, but the software is the same for all models and you just need to pick one with deep memory to make use of the >1Mpts FFT even though this cannot be fast anymore once the resolution bandwidth drops down to a few Hz. But down to some 100Hz bin width you get a quasi real-time display.
I do have 3000 and 4000 series PicoScopes and I'm very pleased with them.
A Siglent SDS1202X-E also has a very nice and fast FFT up to 1Mpts and would be within your budget, yet I've recommended the PicoScope for several reasons:
1. You've asked for the absolutely fastest. That's the PicoScopes, even though the difference to decent & fast benchtop scopes like the SDS1000X-E isn't huge, especially on narrow RBWs.
2. For more sophisticated applications, the Pico SDK together with the continuous streaming capability might be a bonus.
3. The user interface of the Pico FFT is pretty much like a spectrum analyzer, so it's certainly the easiest to use.
4. The PicoScope provides automatic measurements like THD, SFDR, SINAD and more.
Here's an example what a 16bit Pico 4262 can do in analyzing a 19.9kHz sine wave from an AWG and its harmonics up to 1MHz at 1Hz bin width:
Pico4262_2M_19900Hz_Span1M_avg