Well, VPN or remote desktop doesn't make it any cheaper, you still have to have a FileMaker client license. And FM GO is a pretty inexpensive solution. Your client probably has some unrealistic expectations on the costs of a database. But that is another story.
You can use all the normal methods of "screen sharing" from Citrix to Remote Desktop and they have varying security levels. Citirix tends to be the most common enterprise solution, but then again, your client apparently is cheap and suggesting doing things the right way is not the inexpensive way. But if they are cheap, one of the easiest solutions is so screen share with a free service like http://join.me or setting up VNC sharing. By the way, Microsoft's Remote Desktop works just fine on Macs.
My client wants to offer his salespeople access to his database which is running on FMSA 12 .... he wants all the good stuff at a low cost
FMGo is going to be the most cost-effective distributed solution.
The question for them is simple, is it a profitable investment? Put it to them: if they do not expect to obtain reasonable benefits on the investment in their business you should move on. In my experience, good clients will appreciate your work, respect your advice and pay promptly (and the world is full of good clients. They are out there, wondering where you are).