Heya,
The 100-400 Mark II is superior in basically all ways. The weight is a wash frankly in my book (and if that weight difference mattered at all, you'd be better off using a much smaller system like a M43's, etc). The 300mm F4L is a great lens, its just one of those earlier lenses where for a good price you could get F4 speed, 300mm focal length and very fast AF with a sharp wide open high contrast image. Fantastic for close range anything and especially great for larger animals at closer range on a bigger sensor. Ultimately though, its an aged lens. F4 isn't as important as it used to be on this focal length as our tech allows much more sensitivity than it ever did that is useful. So going to F4.5~5~5.6 really isn't much different, and while there's a tiny bit of DOF difference in that for subject isolation (if you care about that at all), it's not enough to really fuss over I think here. And honestly the shorter your focal length and closer you are, the faster the angles are changing if you try to track something, so fast, that you probably won't be able to keep up panning with it. It's significantly easier to track a bird at a vast distance with a long lens, as the angle reference frame is not changing nearly as fast relative to your perspective. So keep this in mind. Its so easy to say "get a short fast lens and get close" until they fly and you can't even keep it in the FOV.

IMAGE LINK: https://flic.kr/p/QedhBE
BirdBlind_1D3_Fuji_Squirt
by
Martin Wise
, on Flickr
I just sold my 70-200 F2.8L, 300 F4L and 150-600mm. I no longer any have any AF long lenses. I don't really track birds in flight anymore. And I much more often just seek out birds in their habitat doing what they do, not flying though. I'm just using my telescopes with way longer focal-lengths on a portable alt-az mount and attaching my APS-C mirrorless camera to it these days. These days I'm doing blinds/hides and perches and I walk some trails and look for birds hanging out. I no longer chase Terns in flight or anything at the coast. I actually just like to watch them more than try to photograph them I find.
IMAGE LINK: https://flic.kr/p/2kMHSnr
ED80_Fuji_XT1_01
by
Martin Wise
, on Flickr
If you need reach and AF, the 100-400 Mk 2 is one of the most versatile lenses to carry. And if you need even more reach, go for the 500~600mm options out there. Of course it depends on what kind of birding you're doing. Big birds at far distance is way different compared to fast moving close range birds in flight!
Very best,