If a DW fact table is not uniquely identified by all dimensions, what problems?
I don't think you will run into a lot of problems when querying your data warehouse except for the one you already stated (cross fact queries) although if you are aware of how your design is done and only aggregate over dimensions I think you should still be able to query that. You couldn't join on the dimension keys in the fact table but you could still aggregate. 

If you only care about aggregations on a daily granularity there is no need to have a dimension with a finer granularity in my opinion.

Where you might run into problems is if you try to do incremental loads or update fact records in your ETL instead of doing a full load but there are ways around that.


You could use a fact table surrogate key as Kimball describes.
You could use the technique described by Vincent Rainardi  
You could include "supporting" columns in your fact table that you never report on but support your ETL strategies (in your case timestamp maybe)