My wife and I fly quite a bit. We routinely through check on flights with a airline change, even if they are not on the same ticket and locator.
Just show your second ticket when checking in for your DL flight. The bag tags will be printed and checked all the way through to Georgetown.
FWIW, we do this when transiting Singapore, Miami, Houston, Detroit, and Seattle, all with no issue.
If you want to, you can deplane in MIA, go outside to baggage claim, pick them up and drag them to your next airline, but it's a PITA supreme.
FWIW, we often transit MIA enroute to Nassau, through-checking, with a airline change just as you describe. Zero problems.
The return is also not an issue. You will have to pass through Miami Immigration after deplaning from your Georgetown flight, then, pick up your luggage and hand carry it past Customs/Agriculture. After passing this, there are baggage conveyor belts for each airline, you just flop the bags down on the belt if you have it checked through. If you did not through check, you will have to find the DL rep and have them print a new tag, a PITA.
Have fun on the trip!
All the best, James