Most of the electricians kicking around this forum will probably cringe--considering this to be "sloppy work"--but, yes, I believe you can use a segment of 12-2 in a 15 amp circuit.

There's a slight chance that an inspector would tag this as unworkmanlike, but it is not inherently unsafe. Perhaps the main reason not to mix wire sizes is the possibility that someone else, down the road, may incorrectly infer that it is a 20A circuit. With that in mind, you might tape on a tag noting "15amp circuit".

Note that Home Depot will cut you a 6' piece of 14-2 for about the price of a BigMac. (Don't forget that you'll need 6" - 8" inside each of the boxes.) It's your call, but unless I was many miles away from a good source of custom-length wire, I'd spring for a short piece of 14-2, if only for pride of workmanship.

There is no problem with using #12 wire in a 15 amp circuit. An inspector would not flag that.