I would make sure to call around to all the local banks in your area (portfolio lenders) and see if they have a seasoning period.  

@Ryan McNamaraMartha’s Vineyard has a very high entry point and a 8-10 week rental season.

Basically, can I directly write a check from my HELOC to fund the down payment at closing or will it need to be seasoned in my bank account?

Have you considered self storage?

We're also planning to add coin-op laundry in the basement, and possibly large storage lockers.

People will use it for a rec room, storage, etc.

I am by far not an expert in this but looked into it a bit in the past when researching building storage units in an AE flood zone.

You will want to make sure to ask the lender what their seasoning requirements are before you can use the new appraised value.

Does anyone use a bank that doesn't require much "seasoning" of the leases on a triplex? 

Be sure to consider properties with year around draw or make sure the seasonal nature can cover the payments in the off seasons.Here in Boise, Idaho; we have such a shortage of properties, I know people that have resorted to renting nightly rentals for months because they couldn't find anything else to rent or buy.