I have only stayed in BDA in August, which is warm and occasional rainy days. I assume it would have cooled off a bit by October.

Staying in BDA is very expensive. The best hotels for location on or near the beach are the Elbow Bay, the Fairmont Southampton and the Soneta (which I think has recently become a Wyndham).

As for apartments, be careful what you take. They tend to be IMHO very expensive and not particulaly high quality. I have stayed somewhere called Dawkins Manor, and know people have stayed at a place called Brightside. I wouldn't particularly recommend either. I do recall there being not a lot of choice in the 2 bedroom range - most places are suites of 1 bedroom.

Be aware that with no car hire, you need to be within walking distance of a shop. This is an absolute must. Choose carefully.

Not sure this is particularly helpful, but I may be able to at least tell you more places to avoid, and can give advice on locations etc.

It is a beautiful island though. And probably the most beautiful beaches - Elbow Bay, Horseshoe Bay, John Smiths Bay.