Suppose there is only one Levi in Shul and he does not want to leave the Shul to wash the Kohain's hands since he will lose the Amein's during the time he is out of the Shul. Should the Levi be compelled to go wash the Kohain? (sources)
This tshuva again gives a perfect answer to the exact question. If there are other leviyim then he need not. Otherwise washing the kohanim's hands comes first. If there are not too many people answering amen then he should stay inside.
The Gemara simply talks about a kohen washing hands. It's the Zohar that says a Levi should do it.
If no Levi is available, the job is given to a firstborn; if no firstborn is available, the kohen just washes his own hands.
I strongly suspect that if the Levi doesn't want to do it, just ignore him and go find a firstborn.
(Conflict-of-interest disclaimer -- I am a non-Levite firstborn, and don't get a lot of kohen-washing business!)