On the last move, after only 2 train pieces remains, is it possible to claim a 3-carriage route, if you have the 3 required color cards but only 2 plastic trains left to place on the board?
No, you cannot claim a route longer than the number of plastic trains you have left.
Claim a Route
The player may claim a route on the board by playing a set of Train Car cards that match the color and length of the route and then placing one of his colored trains on each space of this route...
... When a route is claimed, the player places one of his plastic trains in each of the spaces of the route...