Sounds like a job for the almighty Envelope Distort. Bonus - you don't have to expand the text, so you can still edit the text as a live "Type on a path" object.
The easy option is to use one of the options from
Object > Envelope Distort > Make with Warp menu.
If you want more precise control (here's an example where we really mangle the text)...
Object > Envelope Distort > Make with Mesh, choose a number of rows/columns that suits the shape
2: Adjust the points to taste with the direct selection (
A) tool - make sure the top left 'Envelope' button is set to 'Edit envelope' on the left.
3: To edit the text while keeping the effect, change the top button to Edit Contents (the star on the right)