Answer from Edith Harbaugh:
I'd push it til the absolute last item (after partner meeting, when they're about to issue term sheet). Otherwise you get many tire kickers who'll burn out your best customer***. It's fine to be upfront with the VC (& in fact it might make you look better) "Everyone wants to talk to marqueecustomerZ - where are we in process, and unless it's the absolute last item, my priority is making sure they love our product"
***ProTip I learned later - if a VC is legitimately interested, they will be doing everything they can to backchannel to that customer to get "inside information". VCs would ping our customers looking for references, etc.
I learned this the hard way of thinking my customers were so awesome they'd convince the VC, and instead just fatigued our customers on wishy-washy VCs.