I just read the menu on MDE and see ‘The Earl Sandwich’ is turkey, bacon, lettuce, tomato and mayo. Possibly you could ask them to make that without the turkey?
According to one of the Disney blogs that tracks menus, they got rid of the Best BLT in February of 2022.
They apparently still have the Breakfast BLT, which is seemingly a smaller version. (Maybe a little weird that they call it that - it doesn't seem to have anything typically "breakfast" in it.)
We ordered a BLT in early June. I’m very sure it was on the menu but either way we had one. There’s no adult grilled cheese on the menu but we get one every time with no question. I’m sure if the BLT is not on the menu you can still get one.