You've missed one option: WebDirect. Most of what you're describing is fairly low-demand and WebDirect might be a very good option. It's considerably easier to develop than doing pure web coding (FileMaker does most of the heavy lifting for you), and you can get it up and running fairly quickly.
If you don't want to use that, then I would recommend using CWP. There's no need to set up a MySQL database at all; just use the PHP API. The web hosting piece is included with FileMaker Server.
WebDirect would make light work of this job, provided that you don't expect much traffic.
If you want tight integration with the existing web site then using the PHP API or the XML interface. FileMaker provides a user friendly API for PHP but if you prefer perl/ruby/python/etc you can use the XML interface. It's fairly straightforward.
The only problem with web direct is I would have to purchase user licenses for every customer to be able to connect, and we have about 5,000 customers or so.
Uh ... user licenses for what? It runs in a web browser.
There need concurrent connection licenses, may be he confused it per user license.
Unless all 5,000 users are logging in at the same time, then a far smaller number of concurrent connections should be adequate.
However, it largely depends on the use patterns. If this login is expected to be rolled out for something like, say, an event, and you're expecting a big push when the event is launched, then WebDirect may not be able to handle the load (even if you purchase enough connections). That's why we put in caveats about user load.