If a probe entered a highly elliptical orbit around Uranus after firing its engine (similar to what Cassini did at Saturn), could it use Uranus' moon Titania as a gravity assist to circularize its orbit?
I don't think you can circularise your elliptical orbit by only using Titania's gravity, or at least you can't get into a stable circular orbit.
Let's ignore all other stuff orbiting Uranus and assume a three object problem for simplicity. If you are only using Titania's gravity to enter a circular orbit, at some point your new circular orbit will need to diverge from Titania's enough so that it's gravity won't alter it anymore.
Let's start with a coplanar orbit: as Titania's own orbit is already circular, your new circular orbit would be near syncronous with it, and you would be nearly static as seen from Titania. You'll end up falling onto it.
If you want to orbit in a different plane, you'll still have two near misses each orbit, which will again disturb your orbit.
Adding all other Uranus' satellites will only complicate things further.
You could, however, use Titania to lower your apoapsis and then do a short burn to circularise later on when you are far away from it.
Assuming you approach Titania from the appropriate direction, there is nothing stopping you from using it for a gravity-assist either to accelerate or decelerate.
Actually performing the maneuver is a matter of timing and good aim.
So yes, in principle you can absolutely use Titania to alter your orbit with a little planning.
One caveat is that you will probably need a small amount of fuel/thrust to fine-tune your trajectory, especially after the maneuver, because you don't want to get into a roughly circular orbit that grazes the orbit of Titania and in a few months get deorbited onto Uranus by an unintended grav-assist from the moon.