I think I'd opt for the 300/2.8 II with the pair of TCs. Only thing I'd consider is if the loss in FL means you end up with too few MP for your output format if you crop (e.g. 420mm cropped to the FoV of 500mm). AF tracking will be better, you have a lot of flexibility, too.
Caveat on my advice is that I've no direct experience with either lens (except a few shots taken with a friend's 500/4 MkI).
(my 2 cents)
The new Canon EF300/2.8L IS II USM is maybe one of the best lenses to be combined with Canons new versions III of their TC ´s, and also according to the comparison you already made between 300 II / 500 I in regard of IQ, the IQ will be in favour to the new version of EF300/2.8L IS II USM, which also have the autofocus module optimized, and it give you a much lighter rig, and therefore also more easier to transport, and handholding, and has better flare resistance etc I would also opt the same, as Neuro said, and go for the EF300/2.8L II before choosing an old version with old IS etc.
I have been using the version I of the EF300/2.8L IS USM with both versions-III of the Canons new extenders and I am quite pleased with the results - and having the new version would certainly be even to a degree better even if you use the new extenders with the lens. Other comparisons shows that the EF300/2.8L II is even sharper but not by much than the new EF400/2.8L IS II USM and that goes to the favour for the 300/2.8II when the extenders are used.
But you might need to test this combination first sir!
Also to maybe take into consideration, is that, the MFDs (Minimum Focusing Distances) is 78.7" or 2000mm for the 300 II vs 177.2" or 4500mm for 500/ I. This is important if you are really close to birds for example in a blind or hidden in a vehicle.
I wish you many exciting and great coming shoots!
All the Best!