New bag for setup
I like to use a sling bag for walking around with my DSLR. My current Lowepro SlingShot 202 AW isn't deep enough to support my recently purchased a Canon 6D, 24-70 f/2.8, and 70-200 f/4 IS. It works fine with the 24-70, but the 70-200 is too long when attached to the camera.
I'm looking for a sling type bag that will at least hold the camera, those two lenses, and a flash.
I ended up with a ThinkTank Urban Disguise 35 v2 to fulfil those requirements - after a very, very long period of research...
Some quick background first. My kit consists Canon 7D (similar size to your 6D), Canon 17-55 f/2.8 IS (very similar size to your 24-70), Canon 70-200 f/4 (same size as yours), and a Canon 430EX ii flash.
I similarly wanted lenses to fit in attached, and ideally with hoods attached too.
In short, the ThinkTank UD35 allows you to do this.
It's not the size of a monster - in fact it is quite compact compared to my original expectations. It's beautifully (and I mean really beautifully) made.
Just to clarify, down the centre slot, I can have either my 17-55 or 70-200 attached to the 7D; and it's an easy and comfortable fit. That includes having the hoods on in their forward-facing positions. Whichever is attached, the other will fit down the right slot and with its hood attached facing forward. The left slot is not so tall but accommodates my flash (it's not so tall because the grip-side of the 7D must sit on top of the left slot).
Right now, my UD35 is packed as follows: left slot = flash; centre slot = 7D with 17-55 f/2.8 attached with hood in forward position; right slot = 70-200 f/4 with hood reversed and 85 f/1.8 underneath with hood reversed (without the 85mm in there the 70-200 would fit in with hood facing forward). My BlackRapid strap is attached in the main compartment, too. There's a trillion other hidey-holes with stuff in, including a slot for an iPad to the front, and a slot for a 12" laptop to the back.
Amongst other things, I like that it is not too wide (less bumping into other people in crowded situations); it being tall makes it more manageable than a wide bag IMO; and that the bag is beside me not behind me (good for security).
The strap is comfortable, the zips never snag, and it feels like it's going to last forever. With top zip partially opened, the camera slides in and out effortlessly.
Bags are a personal thing but this is worth a look.