There is an exception (I'm using SQL 2014) when you are only using Insert/Update/Delete on Declared-Tables. These Insert/Update/Delete statements cannot contain an OUTPUT statement. The other restriction is that you are not allowed to do a MERGE, even into a Declared-Table. I broke up my Merge statements, that didn't work, into Insert/Update/Delete statements that did work.
The reason I didn't convert it to a stored-procedure is that the table-function was faster (even without the MERGE) than the stored-procedure. This is despite the stored-procedure allowing me to use Temp-Tables that have statistics. I needed the table-function to be very fast, since it is called 20-K times/day. This table function never updates the database.
I also noticed that the NewId() and RAND() SQL functions are not allowed in a function.