As far I understand things, Emerald is an Irish airline and will therefore operate EI-registered aircraft so they will not be able to operate the domestic flights to/from BHD due to Brexit, unless there is some agreement put in place between ROI and the UK that will facilitate this once the current contract comes to an end in December 2022.
Stobart Air/Ettyl use their EI-registered aircraft at the moment because the current contract was awarded pre-Brexit, however they're in the process of applying for and obtaining a UK AOC and have already started the process of putting aircraft onto the G-register (according to Jethro's, an ATR72 is due to become G-CLUR), so perhaps they might end up operating the Aer Lingus Regional flights to/from BHD on behalf of Emerald; who knows?