Thanks all, for your comments!
Not sure this is necessarily the best choice, but I believe I'll go with the 300mm F4 (money limitations) and the 1.4 teleconverter, which supposedly will work with this lens.
If I understand correctly, 1 stop light loss would be like going from F4 to F5.6.
Since nearly all my wildlife shots are in daylight, I should be okay.
Any other thoughts on something (s) I may be missing?

Hello All!
I'm about to purchase a longer lens for my Nikon D5200.
I'm currently using an AF-S Nikkor 18-200 1:3.5-5.6. In general, I'm pleased with the lens, but I do a lot of wildlife photography (mainly birds) and find that the 200mm doesn't give me enough reach.
My question is, with a limited budget, which would give me the "biggest bang for my buck"----an 80-400mm zoom, or affixed 300mm F4 with a 1.4 teleconverter?
The zoom lens is twice as expensive, but I think using a teleconverter would cost in terms of light.