You don't create software "under a license". You create software, then you own the copyright, and as the copyright owner, you license the software with whatever license you want.
Since you want to use the software with more clients, and your client wants to use it as well, you could just add
Copyright 2017 by user1032531
Unlimited, non-exclusive license granted to TheirCompany
In other words, they can do whatever they like, except that you are the copyright holder. So they can't stop you from doing anything, and they can't sue someone who makes illegal copies of the code (only you can). GPL has restrictions that they might not like.
In many countries, you can't claim you're the author if you are not the author. You may be the copyright owner, but to be the author, you must have written it yourself.
GPL restrictions: If you give GPL licensed software to anyone, you have to give the source code to them at the same time. OR you have to give the source code to anyone (and I mean anyone) who asks for it.
Let's say your client hires me to add to your software. And I don't want my code to end up GPL licensed. Then your client has to find someone else to do the work.