I am not being critical of your suggestion but why would Flybe want to?
They are profitable as a regional operation, and would instantly be up against stiff competition. Crewing would be more costly as well for a Gatwick operation.
If it ain't broke, don't fix it is a mantra often ignored to the cost of many airlines. Flybe are profitable in their niche and Gatwick would be a hefty gamble with that profitability.