We have traveled with passports expiring within the 6 month period before. I actually had never heard that rumor at the time and so never even thought about it. And it was my kids' passport so if ever they wanted to crack down that would have been the time.
As per the "rules", the passport has to be valid for the duration of your stay (duh) but residents must have passports that are valid 3 months past entry.
I would not risk sending the passport away for a new one. Our one passport renewal last year took about 7 weeks with expedited. I cut it very close.
https://www.gov.uk/foreign-travel-advice/british-virgin-islands/entry-requirements