That's a tough one because of course it will be next to impossible for them to "sell" the extra month to another child. So they will lose money if they allow you to register. I'm sure they face this all the time with families in the summer who want to go on vacation, send kids to camp, etc. So they are going to ask for the full 3 months of tuition/fees, even though they know some people won't attend all the time.
And of course, the added problem is that you will be moving so you will no longer be a customer of theirs. So their motivation to bend to your schedule is even less than it would be if they were trying to please an ongoing client.
I think your best best it to be honest: "We are moving but we have so loved being a part of your program that we would love for our child to have a final wonderful experience before we leave the area." Indicate that you know why they have a rule and that you support it-- "We know that you have a business to run, and that it is strengthened by good instructors who are paid out of registration dollars." You are cutting their income by 1/3, so that has to be made up somehow, either by cutting out staff, reducing a coach's pay, or bringing in another kid for the final month (which throws off the dynamics and the instruction).
Your other option is to pay the full 3 months and let them know you will be leaving. Or you can ask to be wait-listed if they don't fill their roster and then would be willing to take your child on a 2-month basis (because 2/3 of their fee is better than no fee at all).
I think you will get more consideration from them if you word it from THEIR perspective and also how much your child will get out of it (based on past experience or on their reputation as a team) than if you take the "I can only afford X" or "I don't want to pay for what I'm not using" approach.