I would definitely not say CTD. And it's not even clear if it should be a UST.
In the past, MBS was hedged with USTs, but then MBS spreads blew out, and everyone realised that USTs could rally and MBS need not do the same. Or MBS could sell off like mad (spread widening) and the USTs might not do a thing. Basically, there has been blood spilled with every major lesson.
MBS is a spread product, so best to hedge with another spread product. So instead, people started using swaps. I could tell you more about how swaps are also a bad hedge, but anyhow....
Most MBS models give you a way for converting from PV to OAS and back again given a specific interest rate model. IR models take as an input swap curves (usually the 1m, 3m, 6m LIBOR swaps and Fed Funds for discounting) and vols. Really, you typically need all this stuff. To hedge, usually, you think of change in PV (keeping OAS fixed) for a given reshaping of the initial swap curve. Basically, the quants will have a way to bump each input swap, so the 2y swap, the 5y, the 7y, the 10y. You will get a delta to each swap.
So, any MBS exposure (even a single bond) will have deltas to each input swap. You'd never hedge them individually, but might if you had a large book.
Intuitively, if you just wanted to hedge with one underlying, it shifts about. The primary rate exposure for MBS is the Commitment rate (FN and FH both publish 30, 60 day commitment rates). The commitment rate is at times related to 10y swaps, at times much shorter, depending on the overall duration of outstanding MBS. It's a bit circular, but in a refi wave, you get a dependence which is closer to the 7y or shorter, and during a sell-off closer to 10y. Other people will model the commitment rate as related to some points on the swap curve with a vol component. I would say, if you really must hedge with one, look at the FH and FN commitment rates and figure out how their dependence on swaps of various durations has shifted through time.