Seems like the Blackrapid strap is the answer for me, but will the mount be able to take the weight? The 70-300L doesn't come with a tripod collar, so should I get one?
I always leave my collar on, I would say it is with buying one, but it doesn't have to be oem... there are some decent knockoffs out there.
The Canon collar for the 70-300L is expensive. There are decent knock-offs (I use the Fotodiox with a Wimberley P20) that work well.
The camera lens mount could probably take the weight (see Neuro's example with the 85mm L), but it is much better balanced on the strap as well as on a tripod if you use the lens collar.
An alternative to a strap can be a shoulder bag that is tall enough to house camera with lens and hood attached - but you won't have the camera ready as quickly, and if you are already carrying a backpack with more gear, it is a little busy around the shoulders. I do it sometimes when I take camera and 1-2 lenses only and leave everything else at home.