My thoughts based on what I own & have owned.
300/400 f/2.8 Version 1 Lenses, both great, but heavy.
300 & 400 f/2.8 Version 2 Lenses, IQ I feel about the same as the Version 1 Lenses, but 30 percent less weight.
600f/4 Version 2 Lens, lightish, great IQ
200-400f/4 great versatility, great sharpness, but horrible price.
When I purchased the 200-400f/4 I sold the 400f/2.8 as I dont feel I'll use the 400f/2.8 that much anymore.
I kept the 300f/2.8 II & 600f/4 II as they do a different job, 300f/2.8 great for low light, hand holdable, all round in my view the sharpest Lens Canon make, works really well with Series III 1.4x
I kept the 600 f/4 as with the 1.4x gives me some Legs.
If you shoot Birds, at some point you will need good IQ 600 +, the 600f/4 with 1.4x will give you that with spades, clearly seen from Images from people such as Gary Samples with his Eagles.
The 300f/2.8 I will work very well with the Series III 1.4x, But for Birds I think you will find yourself short most of the time.
The 2x Converters are in my opinion almost worthless as they degrade the IQ way too much, but they give you an Image at a price in an emergency.
If you shoot mostly wildlife your 300f/2.8 plus your 1.4x converter should work just fine, look to upgrade at some point when you can afford it to the 600f/4 II if you find your shooting BIF more than other wildlife, the issue is you may then find the 600 is too long for wildlife but just fine for BIF, Life, full of difficult decisions.
Which ever way you go, good luck with the Imaging.