Heya,
If your 120-300 was longer, I'd say yes, get it. Having a 400mm sidekick to a 600mm setup for example makes sense. But your 120-300 already does what this lens does, with stabilization, and it's faster focal-ratio. Put a 1.4x TC on your 120-300 and you're set, at F4, stop down to F5.6 just to ensure sharpness and you have the same thing, with stabilization. Sure, it weighs more, but carrying both weighs even more. And you'll miss versatility of your zoom when you are static on just 400mm.
For me, I commonly was shooting a 150-600 for birds on both full frame and APS-C. I got a 300 F4L IS as its sidekick. I now actually use the 300 F4L IS on my 7D more than anything as I found I was getting closer to my subjects and didn't need the 600mm as much. I really enjoy the light weight, size, speed, everything of the smaller shorter lens with stabilization. So for me it made a lot of sense to get that shorter, lighter lens.
Thing is, for you, you already have a 300 F2.8 basically. And you can throw a TC on it.
Instead of a 400 F5.6, I'd just get a good monopod with a tilt head, and keep a 1.4x and 2.0x TC in the bag. You're already set.
But, then again, just to play devil's advocate, if you're really just not enjoying your 120-300 F2.8 OS because of its weight, then maybe you should be looking at a lighter lens. You'll do more shooting if you enjoy the gear. And the best lens is the lens you actually use after all.
You can always get the 400 F5.6L or a 300 F4L IS or something like that, with a 1.4x TC where you want. Try one out. See how it suits you. If you like it, keep it. If you don't, re-sell it. It's like a free rental and you get the experience.
I personally again went with the 300 F4L IS route, because it works well with a 1.4x TC to get me 420mm if I feel I need it, has stabilization, and is F4 and sharp at F4 with a very close (4.9 feet) minimum focus distance, so its a very versatile lens, can do macro pretty much out in the swamp for when I'm right up on something, or I can extend out with the TC. I also shoot on a 7D, and it performs great. Light weight. Stabilization. I chose this setup over the static 400 F5.6 without stabilization for the versatility. It compliments my 150-600mm, but honestly, I often don't even take the 600mm because I can get close enough to not need it these days. I got my 30o F4L IS for about $500 from KEH.
I still will make an argument for having a mount though. Either a tripod with a gimbal, or a monopod with a tilt head. It makes a huge difference with telephotos. So much easier to smoothly pan and track something in flight. And for things that are being still, well, you get a lot more odds to keep it sharp in the wind and it takes the weight off your arms while you wait for the right moment. Highly recommend you try something if you don't already use mounts with long lenses.
+++++++++++++++
I use a monopod + tilt head, and a tripod + gimbal setup. When walking, I do the monopod and tilt head. When sitting or hiding, I use the tripod and gimbal.
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IMG_9890
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Martin Wise
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Martin Wise
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Even in the rain
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IMG_9882
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Martin Wise
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Very best,