I teach a few kids guitar, and the best exercise I know of is to learn Bm and F# together. Which is why I start with Am and E. I get them to learn those two chords smoothly with normal fingering. Then I get them to play Am with fingers 2,3,4 (leaving finger 1 for the barre). Same with E. Then I get them to play the Am - E - Am progression with the new fingerings. Only when they can do that do I get them to play Bm and F#. All they need to do is hold the barre finger in place and practice moving between the Am and E shapes as before.
A song like Hotel California (as noted above) is good for this. So also are blues songs in Bm such as TheThrill is Gone by B B King.
All the best!