The only thing I can think of then is that the LMV385 has some internal issue. It seems to b functioning as is required in the Arduino, but is getting hot. Either it is consuming a lot of current, or something attached to it is consuming a lot of current.
As far as external things go... Since the LED goes off with digital pin 13 grounded, then the problem does not lie on that output. You say the FET measures "okay" in circuit, but that does not mean it is behaving properly when power is applied, and it is the only other external device that could be causing the problem.
So, given that the LMV358 appears to be working as expected, I'd say replace the FET as a first attempt. Before you do that check 2 more things, again with the 328 out of the board:
1)Power the board from the USB cable only, jumper digital pin 13 to ground. The pins on the LMV328 should read as follows;
- Pin 1 ~ 0V
- Pin 2 ~ 3.3V
- Pin 3 ~ 0V
- Pin 4 ~ 0V
- Pin 5 ~ 0V
- Pin 6 ~ 0V
- Pin 7 ~ 0V
- Pin 8 ~ 5V
- Does the 16U2 get hot at all?
Edit: BTW, your measurements of the NCP1117 are what you should see. It appears to be working fine.