The mat should not get hot enough to melt the plastic tank, if used with a thermostat like they should be.

Try get it to a vet as soon as you get back. Even if it's a normal vet they might be able to help or get in contact with a herp vet. If the frog has a calcium deficiency then treatment right away is essential, otherwise if MBD gets to the stages where it cannot be reversed the frog will probably have to get put down