I just did the trip from Nanny Cay Tortola to Ft. Lauderdale to drop our boat off to the new owners. Really an easy trip assuming you have offshore skills and good charts. We passed to the south side of PR
(much calmer) and then up along the west side of T&C, through the Bahama chain and across from the north side of Bimini. Non-stop you can do it in a week and if some serious weather pops up in the forecast you always have a couple days to find a place to hide and that routes provides plenty of emergency stop options if needed and as long as you are checking weather daily there is little chance of "getting caught".
We sailed our monohull over 30,000nm in the past 3 years (just sold her to go to a cat) so we understand what it's like to "get caught" but honestly I would make the trip you propose now if needed. Hurricane forecasting is so good these days you are always (except in the extremely rare situation) going to have 3 days to run and hide. Some will argue that "hiding" is still very risky given the power of a hurricane but in the worse case you would have a day or two to find a spot, secure the boat as best you can, and then get off to a safe area on land. A lot better than being at sea in a cat 2 or more. Of course some risk but there is always risk of one kind or another and in June the odds are in your favor by a long shot. Sure not the ideal time but hey, life is what it is.
From Fl you get a nice gulf stream push and as long as you avoid N wind (again forecast is very good for 2 days) you will make great time.
If however you don't have a good reliable way to get weather info while offshore (SSB or Iridium) don't do it!!
But...don't wait any longer, get going!