just to make it clearer
Publisher
------> Distributor
---------------> subscriber 1
---------------> subscriber 2
---------------> ....
---------------> subscriber N

the above can be any server - they could...

why not just convert the date to a string on the source select - converting to a format that SSIS knows well will then make the conversion to date straightforward. 

Using a linked server can also be done through synonyms - so they do fall under the same as a hard-coded db other than the code that is referencing them.

A few comments on your code.

Do not as a standard store user functions on master  -- master.dbo.ufnStringToPbase  should be on any db other than system db's
Do not...

I do think you took the wrong approach there unless you didn't tell us of further steps that would normally be required.
But by your list of steps I...

the .net version of SQL Server has nothing to do with what your code does. Unless its a CLR, and even that I think it will still allow it.

by the error you posted it seems you tried to use a format that is not possible
Create failed for Login '.\TestAccount'.
drop that .\ and it will work

in my shop I try to implement and force the use of synonyms. pitty I can't force them to do it (not being a production DBA), but do try my...

good to see possibilities 😀 and I didn't knew about it either!!!

powershell definitely an option - and possibly plenty of examples on net to do it.

this may also be an option https://blogs.msdn.microsoft.com/markm/2015/06/30/easily-backup-sql-databases-to-azure/