I have a little experience with the OP's question, so here goes my 2 cents worth.
First I use both the Canon 1D MarkIII and 1Ds Mark III bodies. I use my 1D Mark III exclusively for wildlife, BIF's, etc.
I have rented the EF 400mm f/4 DO IS USM twice, and the 400 f/2.8 IS once. I wanted to make sure before purchasing a true super telephoto.
The 400 DO is a terrific lens, not in the same league as the 400 f/2.8, but then it does not weigh anywhere near the same either. The 400 DO can be comfortably hand held for extended periods of time. The image quality without an extender is very good. I found the image quality degraded with the use of tele extenders though.
In my opinion the 300 f/2.8 is the "mack daddy" of the Canon super telephoto lenses. This lens is terrific! It takes tc's well, even the 2X. Focus speed. low light use .... awesome lens. It is however a little short on reach for smaller birds, etc. it's also a heavy lens, so some folks do not care for it based on that issue alone.
I ended up with the 300 f/2.8 IS and the 500 f/4 IS as my primary long lenses, with the 100-400 as a kind of a fill in lens.
Having rambled all over the place let me get back to the specific question;
When considering the 300 f/2.8 versus the 400 DO IF the major consideration is "hand holdability" then the 400 wins easily. You don't mention what your primary photo subject is that requires the additional reach, so I am guessing or being general here.
For simple pure reach get the 400. For a lens that is superior in my opinion and takes tc's better get the 300. I personally feel that the 400 DO is overpriced especially when compared to the 500f4.
One thing that I wondered about when debating on the 400 DO is why it was not designated an "L" lens? You would think for a lens that costs about $5,500.00 new it would have an "L" tag. Something else to consider is the resale value .... will the 400 DO hold up in value? I don't see many that come up for sale on the typical sellers forums, so it is hard to judge that issue for now.
Just my humble opinion .... hope that it was of some help.