Why do you want to use a MOSFET, rather than a BJT?
The only advantage I can think of is, generally MOSFETs have a wider safe operating area, than BJTs, but this isn't a factor, unless it needs to drop a large voltage, say >40V.
As mentioned above, the main issue is a MOSFET has higher threshold voltage, than a BJT, so it will have a much higher drop-out voltage, than if this circuit were made with a BJT.
Yes, this circuit does compromise the LM317's over-current protection, but there are ways to fix that. How much current are you planning to draw from the output?