Must the Exultet be sung by a cleric?
Short and simple.
The short answer is: It ought to be.
In the Novus Ordo I believe it is permitted for a non-cleric to sing the Exsultet. I guess that option looks more to the aesthetics of the moment than to the true dignity of the laity and the proper role of the clergy.
In the traditional Roman Rite, the singer must be a deacon or priest. A priest would vest in the dalmatic.
Come to think of it, a bishop could do it. That would be interesting.
The Exsultet, also spelled Exultet, is the liturgical year’s moment for the deacon, par excellence.