The most sensible option would be to create some kind of overall brand that then franchises down to the likes of BI and LM with both companies working together to build a sustainable regional network. I cant see Eastern joining this party purely for the historic run ins they've had with LM and continue to do so with the competing routes. LM and Eastern are the equivalent of Blue Islands and Aurigny, anyone familiar with the channel islands knows the huge amount of 'bad blood' between the two companies.
At least in the short term once restrictions are eased, for passengers there will be a great amount of choice and in theory competitive prices, it may end up being a battle of who's pockets are deeper... didnt Eastern recently get bought back by its original owner for a very low sum?