I kind of chuckle when I hear that a camera or lens is "heavy".
It is all relative, I would not have purchased most of my lenses if I listened strictly to forum type reviews or online type reviews. Testing a lens on one's camera or testing a camera oneself is basically the best way to make a good decision for oneself. When I put the 70-200 on my 7D (in the store), I was very surprised how the balance/weight seemed to help me be able to steady the lens.
I carry the 7D/70-200mii and the 5d/24-70 on almost every trip I make. There are times while out that I change lens while shooting, but the bulk of my shooting is with the above mentioned combos.
One of the other real important issues, is learning how to shoot with whatever combo you may be using. Sometimes it takes more than one trip out to finess the combo. Like any other endeavor one undertakes, if you want to be good, you need to invest time and educate yourself.
Rent both lenses (hopefully you will have a good copy of both) and spend a couple of days shooting with comparison of lighting that will be identical with both lenses.