Is the boulder on the peak of Tycho Crater the core of the impactor, or is it a random rock?
It is just a rock.
A complex crater like Tycho is formed in several stages as the rock behaves like a fluid. The initial impact completely destroys the impactor and excavates a large cavity in the moon, and ejecta is shot out to all sides.
Powerful shock waves push the rock to the side building large walls. As these push down on the edges of the crater, the centre is pushed up forming a central peak. Further faulting and cracking will continue to modify the crater and the peak.
It is possible that this is part of the breccia that falls back to the surface having been shot upwards in the initial impact, but that seems very unlikely. It is more probable that this is part of the ridges of the central peak that cracked and fell into the shallow valley on the top of the central peak, which formed during the faulting as the peak was pushed up from the crater floor. In all likelyhood, this is a piece of moon rock. As it hasn't been exposed to solar radiation for billions of years, it is comparatively light. It can't be a piece of the impactor, as that would have been completely destroyed