I think that a sort name like "Taildraggers, Too Slim and the" could be useful. Think of a record shop, how and under what letter it will sort the band, probably under T but not of Too, but of what they are: the Taildraggers. And then in this case is added to the band "Too Slim" and if that is a featured artist the sort name also could be "Slim, Too and the Taildraggers"
For the Beatles the same: people are not looking for artists named "The" but for artists named "Beatles" so that's why the sort name is: "Beatles, The" and at the record shop it will be at the letter B.
This one you also can sort in two ways, depending which artist you want to feature: Norma Beasler or The Rosebud Orchestra. In the first case, featuring Norma Beasler, you could give the sort name "Beasler, Norma & The Rosebud Orchestra". In the second case, featuring The Rosebud Orchestra, you could give the sort name: "Rosebud Orchestra, Norma Beasler & The".
Hope this helps
UPDATE: edited in "Beasler, Norma and The Rosebud Orchestra" the word "and" into "&" as it was a typo but there seems to be discussion about it, so better correct it! Though it had no substantive meaning at all, using "and" instead of "&"