Should I put indefinite article in the sentence She became (a) soloist of/with the American Ballet Theatre?
I'd say She became a soloist at the American Ballet Theatre.
It actually depends on the condition and number of positions.
If the American ballet theater has only one "soloist" position, then you should say "She became the soloist of the American ballet theater."
If the theater has many positions as a soloist, and she was given one of those positions, then you should say "She became a soloist at the American ballet theater."