I think normally people need a methodology to decide what focal length they need. Because the only one that is on the field, literally, is you.
1. Focal length
As you already have a 200mm lens, you have the starting reference point.
Take one of the typical photos you like, taken at 200mm.
Then, to imagine how it would look at 400mm at the same photographer's position, just reframe it and crop it 200% or 2x, and now you have the 400mm framing.
For a 600mm do the same 3x.
This is not an exact real-life result, because construction and design of the lens, but will give you a good estimate.
2. Shutter speed, noise, and aperture
Then you have to determine the speed of the lens.
Determine the aperture and shutter speed of your photos. Do you like the blur? Do you expect less motion blur? Are you pushing too much the ISO to get that speed? Do you accept that level of noise? Can you push the ISO 1 step more?
An old "rule of thumb" is to have a shutter speed "equivalent" of the focal length to avoid hand-held camera shake. Although you most likely will use a tripod, it is an interesting starting point.
So, can your camera shoot at 1/400 s with a decent amount of noise, in your given light scenario? What aperture are you using? Do you need a faster shutter speed?
So now with the info you have, you can determine if a 2x teleconverter is suitable for you. or a specific lens with specific apertures.
3. Sharpness and pricing
For that extra sharpness, do you want to pay the extra $$? Can you live with less sharp images? Can they be improved using software?
Are you really using the full file for a large print? Or are you only using the files for social media, and then some cropping is enough?
4. Extra considerations
Do you need an extra bag? Is the combo portable enough? Will you be using more a 150-600 and leaving the other at home?
Do these tests, make notes and you will learn a lot more than you expect!