Can I pull 2 amps through 22 AWG stranded wire?
Let's say the wire is 30 cm long because there is a per-foot value for resistance. The table in the Wikipedia article says that will have a resistance of 16 mΩ.

Let's calculate the power dissipation: (2 A) * (2 A) * (0.016 Ω) = 0.064 W.

You've got two wires carrying the current (there and back), and they are a bit less than 30 cm long, so the maximum total dissipation will be about 0.1 W.

There will be no problem. (Also, I am not a qualified electrical engineer.)