I just checked the datasheet for the LP2950cz. I agree with your assumption that it may be the problem. It is clearly low dropout, but it also shuts down at about 200 mA and has very little current capacity (100 mA) and heat dissipation in the TO-92 package.

I would try a standard LM7805 or similar VR, even a standard LDO and see what happens. With the 7805, you will need more head room voltage for the battery, but at least you can localize the problem. Another possibility is that your VR is oscillating. Do you have a scope to check it? Note, the pinout for the 7805 is different than the LP2950cz.

John