The irony is that the existing Thameslink route used to be run by GoVia, and so was Southern so it would have made sense to merge them then.
What I would do in future is farm off some of the Southern routes not associated with Thameslink to South Eastern, rename South Eastern as Southern (as it was once all part of the BR Southern Region). This might make South Eastern too big, so I would then bring back the WAGN franchise (or at least the West Anglia part) and merge that with the North Kent services currently run by South Eastern to form a new franchise, which would be known as West Anglia North Kent. Passengers living in that area could then be known as West Anglia North Kenters or for short, .......

What I would do in future is farm off some of the Southern routes not associated with Thameslink to South Eastern, rename South Eastern as Southern (as it was once all part of the BR Southern Region). This might make South Eastern too big, so I would then bring back the WAGN franchise (or at least the West Anglia part) and merge that with the North Kent services currently run by South Eastern to form a new franchise, which would be known as West Anglia North Kent. Passengers living in that area could then be known as West Anglia North Kenters or for short, .......