I personally don't know of any way to just mail an election to the IRS.  This the IRS's list of where to file statements and forms and the 172 election isn't on there.  https://www.irs.gov/filing/where-to-file-certain-elections-statements-returns-and-other-documents

I think I would just explain it to the client and send it in as an amended return. 

Also, I don't know about the other Intuit softwares, but if you are using Proseries, you don't have to attach a PDF.  You can make the election by going to Forms>MISCELLANEOUS>Part II and select 2. Elections Statements and then check the box on Line 9 for 172(b)(3).

Hope that helps.  Good luck!