I would not depend on ATMs.
Cash YES
Kitty yes (Boat expenses including beer/rum/etc... Adjust if non drinkers. My experience is some people will be cognizant of their "fine wine" habits and will purchase it with own money, not Kitty$$$). I agree for 1 week, including initial groceries, $400-500 per couple. I too would rather split the remainder than re-contribute. Also on last night, if dining out, can help pay tab, assuming all have similar habits. Some adjustment for non-drinkers and kids, but simpler to pretty much keep all boat expenses coming out of the kitty.
The initial grocery/ supplies generally are about $600 at Riteway for us and we use credit card online. Not worried about using CC at resorts(Leverick Bay, Foxxy's, Sydney's etc) However, have been there when no electricity on the island and without cash you may be thirsty.
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Regarding notifying Credit Card Company, (per Capitol One)with the new chip cards, no need to notify.

Always good to have a spare as even in USA, CC can be compromised... Having to have one Fed-Ex'd to you in BVI not convenient