I'm just curious why the client wants it recorded.....

You need to be careful that their intent is not to have you play the engagement one time, then play back your recording for future events. Don't laugh... it happens!

If you need to go out and buy equipment, and spend time to make sure this gets recorded well, you should probably be in a position to ask something from the client as recompense. Possibly (if you can get them to admit they want to use it for entertainment purposes) even enough to cover the cost of the recorder, if inexpensive. Why should YOU lose money over this?

And, finally, there's nothing wrong with a good old fashioned cassette recorder if you get your levels right....!