My company kinda has the same type of deal. They will put you up in a hotel, pay for dinner and entertainment, if and only if, you can prove to them that spending that extra day (or two if you can swing it), will substantionally lower your transportation costs.

For example, I too was asked to go to Miami for a client meeting. This was on Fri morning I was asked. On Tues afternoon I was on a plane to MIA. As you might see, the airfare was out of control. Add a Sat stay and it droped signifigantly.

I ended up making it a long weekend (with them paying!)