LeisureFirst

It's not as bad as you seem to think. The flight is all on the same PNR (otherwise there would not be an issue with your taking a different way of getting to London and just using the second flight). So if you miss your connection through no fault of yours, it is BA's responsibility to get you to your destination. Unfortunately Monday seems to be one of only two days of the week with just one flight to Barbados (most days have two) so this would mean flying you out a day late. But you wouldn't lose your vacation completely.

This is annoying, but this could go wrong even if you were flying LHR-BGI with, say, a two-hour connection time. Flights can be late. I always avoid same-day short-haul to long-haul connections unless absolutely unavoidable and then try to leave a huge gap because of exactly this issue.

So, given you've got this tight connection, the only thing you can do is figure out the best way to get from LHR to LGW and try to make it. You should also make sure that the crew on the DUB-LHR flight know of a tight connection and let you off ahead of other passengers (try to get seats as far forward as possible anyway). This becomes a very reasonable request indeed if the plane is a bit delayed. And make sure you tell them when you check your bags in to try to get them priority handling (which they should get anyway if it's a CW ticket). And if you don't make it, then BA should put you on the Tuesday flight.

Your other proposed solutions, such as getting to London by another means and trying to get BA to put a note in your booking, simply will not work.