I think I've figured out how best to handle this. Sometimes it takes writing a forum post to sort yourself out.
Will post solution if it works as expected.
To solve this particular problem, I made my Monthly Budgeted Amount a "ToDate" sum of budgeted appointments at the Appointment Date > Appointment Month level.
Given that the monthly budget amount returns for the first day of the month only, a "ToDate" function will always return the monthly amount, regardless of which day of the month I filter on.
More testing to do at various levels of aggregation, and I need to inspect the SQL for good measure, but so far it is working.
Well, never mind... this doesn't quite work.
If I report at the month level but filter at the day level--even if I don't include the DAY in the output--OBIEE is summing "month to date" budget counts across all days of the month.
So this approach does not work as expected.
Will look at level-based measures again.