As a speaker of American English, I would really only say "Perform" because this is the exact action that is being done when one presents a character in a play as an actor. When starting a monologue, I was always taught to say, "I will be portraying the role of ___" or "I will be presenting a monologue from the play _____". You could also say "I will be acting in a Shakespeare play", but it sounds a bit odd to say "I will be acting a Shakespearean Play", because you are acting a character, not the play itself. I agree with your intuition that "play" doesn't seem to work here. However, this might be due to just my own dialect. In Shakespeare's language, an actor is a "player". You also would say "I am playing the part of _____" Play means something more similar to "portray" rather than "act" in general, but is also more commonly used to refer to the actual piece. This, I think is dialect-specific and speakers of British English may have different intuitions about this word. So, in summary, you can act IN a scene, perform a scene, or play A CHARACTER in a scene. Please let me know if you have any questions.

Perform or play sound most natural but act still works.