Does Express store any statistics to indicate when performance is held up due to being Express?
SQL Server isn't going to tell you specifically that there is a performance limitation due to it being Express Edition.  But in general, if you start collecting and monitoring wait statistics, you'll see whether there are any bottlenecks due to insufficient CPU or memory.  So if you've given SQL Server Express the max number of CPUs and memory, then bottlenecks in these areas generally indicate that it's being limited by the limitations of the edition.  Technically, you could give it faster processors to help overcome the CPU limiation, but I assume that's not an option.

See SQL Server Wait Statistics: Tell me where it hurts to get started understanding what you need to be monitoring.