How do I stop refresh conflicts between Proactive Caching of facts and timed refresh of Dimensions in a SQL Server Analysis Service 2005 OLAP cube?
Sounds like you're getting orphaned fact rows a.k.a. early-arriving facts, meaning you have facts records that don't yet have a corresponding dimension member in one of the dimensions.

There's not really a way to solve this with the approach you're using because it's sounds like you'll always been a in  situation where your dimensions are lagging your facts. But you can work around it by changing your dimension key error options in your fact processing. There is an option to ignore dimension keys not found. What you want to do is set that to "Ignore" and you might have to play around with some of the error limit options as well.

This should do the trick. Not the prettiest of solutions, but it will work and prevent these outages.