No.
I'm pretty sure no government or service renames ships
during their operation. I've heard of ships being renamed after selling them, or after having been captured, but never in the middle of its
use.
Far easier to think that the E-E was going to be named something else than to think the E-D was going to be renamed.
Frankly, I don't think the characters in-universe are as attached to the name
Enterprise as we, the fans, are. Why should Starfleet care what they call their best ship? The E-E could've been named USS
Pansy and it would've still been a top-of-the-line ship.