It sounds like what you're wanting to do is relatively simple, replace a call to a MySQL DB, with a call to an FM DB, is that correct...? If so, then the book you mention above should provide you with the stepping stones you need.
How far have you got...? have you been able to connect to the database...? to perform a simple 'SELECT' like operation...? Where are you stuck...??
Having the PHP files which will communicate with FM on the remote hosted server is possible, however you'll want to try it out and see what the latency is like - you may find it better to hand the logging in over to your FM WPE server, then return the user to the remotely hosted pages for the remainder of the content delivery.
Alternatively, if all you're wanting to do on the FM server is the authentication, then creating a simple JSON (or XML if you prefer) based API on your FM server might also be a good solution - in that way you can perform a back-end query from your current pages to the FM server passing only tiny amounts of data.
Don't forget to ensure that passwords are encrypted when stored in your database and when being passed from server-to-server, either via https, or using a custom encryption system to prevent people being able to 'sniff' username/password combinations...
If you'd like someone to do this development for you, then FM-web integration is what we specialise in, and I'd be more than happy to discuss your requirements and provide a cost estimate for you.