I will speak with my immediate supervisor about this further, but I was curious what a polite or politically correct way to approach this is?
Yes, a meeting with your supervisor is surely needed.
However, instead of phrasing it like "I won't/can't go" approach the situation by exposing your difficulties planning the trip (your pets, medical condition, etc.), so you can then work out how to coordinate and overcome them together.
Try to have this meeting as soon as you can, and consider starting with something like this:
Hey Mr. Supervisor. I need to coordinate and plan on the possible upcoming trip with Client X, as I have some concerns about leaving, like ...[list and expose all your doubts and difficulties here]... I want to have your feedback and discuss how we can achieve a solution which would allow us to work closely with Client X.
Chances are that you will come up with a compromise and a solution that works for both parties. Be sure to prepare in advance and know exactly what difficulties you need to sort out, as well as what alternatives you propose or what are you willing to commit to make this work.
Edit: Seems you are really not willing to go. In that case, I still suggest you approach it in a more constructive way, by exposing the difficulties and issues you have on going.
Be sure to have your medical letter indicating your condition ready, so you can back up your claims.
Also, I'd suggest that you update your resume and start considering other job options, in case this doesn't seem to reach a consensus or compromise that works for both or if they start pushing you or "forcing" you to go.