It may be prudent to draft a new routine if you can, one that takes into consideration both of your pokémon’s concerns or avoids putting them in such situations altogether. This may be the best option, as it seems that your pokémon get along except for this, meaning a compromise and creating a routine they both can agree on may be easier to accomplish than you might think.
Alternatively, if the squabbling is actually grave and affecting their relationship—or if it’s too late to draft a new routine—consider working on your pokémon’s cooperation via team building exercises. Require them to perform tasks together off the contest stage, and try using them in non-contest double battles. The more you can drive them to work together outside of the contest hall, the more they’ll be forced to put aside their differences or otherwise reconcile enough to complete each task.
Also, work with them off the battlefield as a mediator. Guide them into learning how to reach compromises and settle their arguments without spending considerable amounts of time squabbling.
It will most definitely take a lot of patience on your end, anonymous, but with a gentle but firm hand guiding them, you should be able to help them work something out.
Best of luck!