Video tripod is not a bad shout.
Only if you get the tripod ring for your lens. So you can use the body in portrait (video heads don't have a portrait position)
I can recommend the 501HDV as a relatively inexpensive reasonably smooth head, the mvh502 is probably better, and a sachtler ACE is best of all.
The key is counterbalance. You need to have counterbalance to make camera movements very smooth, otherwise you are fighting against gravity and imbalance.
A video monopod could be an option, interested to hear what settings you are using that a monopod isn't stable enough for, must be pretty low shutter speeds?
I tend to shoot at 1/125 most of the time. I'll go lower down to around 1/80 if I'm on a shorter lens but for the 70 200 I'm mostly shooting 1/125. Regarding the tripod ring mount, I use a flash bracket that lets me turn the camera vertical so no need for that other than some extra balance which would be nice. It also helps with redeye. I've honestly never actually shot with anything other than an image stabilized telephoto lens. Either the 70 200 f/4 IS or 2.8 IS. I just rent these though and have been thinking of buying the 70 200 f/4 non IS. So I don't know for sure if maybe I would be fine with a monopod at that speed. I've done some informal testing during the rehearsals to see what it looks like without IS turned on and it seems like they tend to be a bit more blurry but I haven't tested out too much. My other reason for looking into a tripod is I'd like to get where I can have one camera setup on the tripod with another at my side so I can switch back and forth. With the monopod, I'm always having to hold the camera which would prevent this as well. I eventually plan to get the 70 200 2.8 IS but that's too much money to put in right now and I don't want to buy the 70 200 f/4 IS knowing that I'm going to want to upgrade to the 2.8 IS in the future. Thus the non IS seems like a good investment and will be nice to have even after I have the 2.8 IS since it's less than half the weight for those times where I don't need the highest IQ or IS or 2.8 aperture.