The one BIGDODGETRAIN linked to was a large scale (g-scale, garden, etc), so I am guessing that's not the set you mean.
If it's this set, I think the answer is no. The LionChief sets apply a constant 18V to the track similar to TMCC or Legacy. The remote for the engine tells the engine how fast to go. The hand car would not have any way to regulate it's speed, it would see the 18V and go very very fast.
The good news is, you don't need much of a transformer to run a hand car. Any small AC transformer could be used instead of the brick that came with your set to control the handcar.
While it's not difficult, you need to be able to hook up the new transformer using a set of wires with the connectors that slip over the terminals that are under the FasTrack.
This would be instead of the power pack that came with your set. It's important to not hook up both at the same time!
-Dave