Weight is a definite concern with the 2.8ii and was probably the main reason I chose the f4IS over it. I just couldnt stand carrying that thing around all day. I'm much happier using the lighter zoom and filling in with fast primes when needed.
I think with the 5D3 the low light performance is good enough that f4 will fit the bill just fine.
Also you dont want too shallow a DOF in events and gatherings...and i assume you will have a prime for really shallow work)
70-200L f/4 is a highly regarded zoom in its own right and weighs about half the larger sibling.
While I fully recognize (and own) the f/2.8 II as a high quality zoom, I do not agree that it is called for at every turn just because it is a super duper zoom that also does window and vacuums your floors! There is a cult like following that suggests this zoom as a panacea for everything. If weight is a factor, then by all means stick to the f4 and it is a kick ass zoom.