[Alan Anderson]

Short answer: yes, you will be fine if you solder/splice/connect the 22 gauge wire to the 18 gauge wire from the 20A Power Distribution circuit.

It isn't really illegal to have smaller wires in a custom circuit than the circuit breaker can protect. The letter of the rules permits it, the inspectors expect and accept it, and I'm pretty sure even the Q&A has said it's okay (though I can't find it at the moment, sorry).