My favorite BIF lens is an EF 300mm f/4L IS USM. It's easily hand held (under 1200 grams... or close to half the weight of the 2100 gram 400 DO II or 2350 gram 300mm f/2.8 II). In fact, I use it enough that I have two of them!
Above was shot with full frame, which I wouldn't normally do for BIF. But I was shooting some scenic shots when I noticed the redtail hawk hunting nearby and quickly switched lenses to try to get a few images (turned out I was able to photograph the bird for around 45 minutes as it repeatedly hunted around a field). I also wouldn't normally use 1/8000 shutter speed, but the bird went from shade to full sun far faster than I could change my camera settings!
I like using the 300mm for it's size and weight... sometimes use it with a 1.4X, too (I use the Canon 1.4X II Extender).
That combo gives me the equivalent of 420mm f/5.6... with image stabilization
. The EF 400mm f/5.6L USM is a great lens and popular for BIF, too... but it lacks IS.
I haven't used any of the 150-600s, so can't compare directly. I do use the EF 100-400 II and have to say they're similar. The 100-400mm II is heavier and bigger, so I more often use it on a monopod or tripod w/gimbal. The 300/4 I almost exclusively use hand held. That's one of the earliest Canon lenses that got IS and has the type that needs to be turned off at the switch, if the lens is locked down on a tripod. In comparison, the 100-400 II uses the most advanced type of IS, much like what is used in the various super telephotos. The 100-400 II is probably the least expensive Canon lens with "Mode 3 - Instantaneous IS". It's also the modern type that's self-detecting and shuts itself off when on a tripod and gives about a stop more assistance than the earlier type.
Like many of Canon's telephotos the 100-400 II uses a fluorite element, while the 300mm f/4L is one of the few that doesn't (the 400/5.6L and 70-200/2.8 without IS are the only other two that don't use it, that I can think of. FL reduces chromatic aberrations and makes the lens extremely sharp. I gotta say, though, that the 300/4 ain't no slouch. AFAIK, the DO lenses don't use fluorite, either (but prob. don't need it).
I also don't regularly use either of the 400mm DO... have only toyed around with the original a little and don't claim to be an expert on either. I do know that the II works much better with teleconverters than the original did. Many users of the original would not use it with any teleconverter, found the IQ took too much of a hit. The II also got improved IS. Canon only rated the original DO lens' IS for two stops worth of assistance... while the II's is rated for up to 4 stops (note: exactly how many stops improvement depends a lot upon the user).
Where I would expect the 400 DO II to excel is the background blur it renders. The 300/4 and 100-400 II can make for slightly coarse background blur in certain situations, even wide open. Not sure about the original 400 DO, but I know the II has a 9-blade, curved aperture. I imagine the DO lenses can render bokeh close to what my 300mm f/2.8 and 500mm f/4 can do. The backgrounds with them can be stunning... though I have to be careful to have adequate depth of field.
ISO 6400? I shoot a lot of indoor sports and use my 7DIIs at ISO 6400 and 8000 fairly often. I sometimes even use 12800 or 16000, though those usually require some additional work in post-processing. I mostly use an Imagenomic Noiseware Photoshop plug-in noise reduction plug-in. I've made 8x12 and 11x14" prints from those images, with little trouble. Probably could make larger with the 6400 and 8000 ISO images.
Following is a test shot done at with one of my 7DIIs at ISO 16000....
As you can see in the enlarged detail, there is some noise. But I think it's pretty well controlled and that detail is FAR more enlarged and magnified than I'd ever print the image and view it from 18 or 20", the way we're doing with our computer monitors. Image was shot RAW with care to avoid underexposure, lighting was by a single 60 watt bulb about 10 feet from the subject and a small window about 8 feet away. Lens was EF 100-400mm II. Image was ONLY post-processed through Lightroom with default noise and sharpening settings. Slight boost of contrast and curves adjustment, but nothing else and not much cropping (just from 4:3 to 5:4 aspect ratio).
My point is that a change in workflow and/or different post-processing software which allows you to use higher ISO may be as good or even better solution than a different lens. I don't know your workflow or what you're looking for in your end results and would find acceptable, though. Only you can say.
Besides, it's probably more fun to get a new lens!