I opened a 'New Query' window in SSMS. I pasted the following code (well this is a simplified version but throws the same error as the real version).

Create function dbo.GetFull (@dbName varchar(100))
Returns double
As  --   <------------- underlined in red. 
Begin
	Declare @PercentFull double
	select @PercentFull = 0
	
	Return @PercentFull
End 
go

Seems I can avoid the syntax error by switching to decimal.  What's wrong with double?

"Double" isn't a valid Transact-SQL data type, you have to use a different one, see Data Types (Transact-SQL)

In T-SQl float and real represent approximate numerics 

Whereas Decimal and Numeric represent exact numeric data with predefined precision and scale values

Try

Create function dbo.GetFull (@dbName varchar(100))
Returns double
WITH EXECUTE AS CALLER
As
Begin
	Declare @PercentFull double
	select @PercentFull = 0
	
	Return @PercentFull
End 
go

Try

Create function dbo.GetFull (@dbName varchar(100))
Returns double
WITH EXECUTE AS CALLER
As
Begin
	Declare @PercentFull double
	select @PercentFull = 0
	
	Return @PercentFull
End 
go