Why can't the functional integral be derived in a mathematically rigorous way? What are the obstacles that we have to overcome in order to achieve that goal?
In many cases, the functional integral can be derived in a mathematically rigorous way, see http://physics.stackexchange.com/a/29558/7924
The obstacles are in the lack of sufficiently sharp functional analytic tools in infinite dimensions that would allow one to bound the errors in the approximations that physicists use heuristically.
Overcoming the obstacles probably requires to find a way to perform rigorous nonperturbative renormalization of things that can currently be done rigorous only in the sense of formal power series.