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After more than 13 years, and tens of thousands of questions, Moz Q&A closed on 12th December 2024. Whilst we’re not completely removing the content - many posts will still be possible to view - we have locked both new posts and new replies. More details here.
I'd suggest sticking to creating a new email account yourself or from your customer. Its easier for you to compartmentalise what is happening, especially when there is a flood of links!!
I think it is also ethically more correct to use a customer's email address, you after all creating the links for them, not for yourself.
A big drawback for using Google URL Shortener for anything is that it is not private. Simply add '.info' to the end of the goo.gl url and you ahve all the juicy data you need.