I've been thinking about how to answer this. And I think I may have an answer.
I'll be real, it's a bit headcanonish but here goes.
https://www.youtube.com/watch?v=pXCKqt4bWDE
In this video, the doctor MC talks to says "It took three months, and the biggest Starship the UNSC could thow at it just to move it here"
There are some problems with this statement. 1: The UNSC's biggest starship is the
UNSC Infinity. The Infinity was covertly constructed for like 20 something years using scavenged Forerunner technology, Meaning they should be able to somewhat interface with the technology. But when the Doctor says this, we don't know exactly what she means.
The Infinity is probably the UNSC's greatest asset. And at the time the Composer was discovered, I strongly doubt that they knew what it was capable of, and the UNSC Infinity was likely on a mission anyways. So we can't really use this statement for anything because it's purely speculative. And we don't know if the Composer was moved before or after the Infinity was even deployed.
The Didact being a Forerunner is of course able to interface with Forerunner technology and make the Composer respond and move. Considering what it's capable of, I find it hard to believe it would be any less capable than the Didact's cryptum when it comes to self movement.
"But why did the Didact need the Mantle's Approach to move it?"
Probably because he needed to safely transport it. After all, even scaling its durability via Newton's Third Law. A combined MAC assault from some moderate UNSC ships could probably destroy it. While the Didact can very likely open slipspace directly like the Warden Eternal, and has demonstrated surviving it. Never mind the fact that Gen 2 Mjolnir can as well. He probably can't travel really far interstellar distances really quickly like an actual starship could.
Another theory I have is that he needed the Mantle in order to give the Composer enough power. It was stated somewhere in Halo 4 that the Composer can fire on its own and without being attached to anything. As it did so to an unnamed research team. But when attached to the Didact's ship, it's what allows him to have the massive AOE that he has.
My conclusion: The Didact can likely have the Composer hover high above him like his Cryuptum can. But it would likely be severely nerfed in terms of the range it can cover. Being generous I'd say an area os 2-3 kilometers considering it Composed everyone onboard the station.
So yeah. The Composer's ability to move and hover should be comparable to the Didact's cryptum which is both bigger and likely not as powerful.