So it doesn't appear that the LM317 peak current ability is the issue then. Sounds more like a peak voltage issue.
Diodes are unique cause its hard to use ohms law on them. V=IR , there isn't really a known R for diodes. Consequently, one normally uses a PIV graph or specification. Meaning at some current I, you should have a voltage drop of V. So, if you cannot provide the voltage required for a given I, you are then limited to what current is associated with your peak V. Any chance you know what the expected peak voltage drop across a M140 diode is at 1.5A? (sorry, i've been gone from lasers for awhile so thats a new diode for me)
A A140 diode needed 4.45V in order to be driven at 1.5A according to this thread.
http://laserpointerforums.com/f65/post-your-ma-mw-51432-10.html#post728480 btw, this is all assuming you have good heat sinking of the diode.
with 2x18650, i'd assume you would have enough voltage but perhaps not? Its important to take into account possible voltage dropout. So, at 1.5A, many LM317s have a dropout of around 2.5V. Then you got your series resistor of 0.33 ohms * 1.5A = 0.495 V. In addition to all this, i'm guessing you got cheap multimeter which probably has a crummy internal sense resistor which provides an additional source of dropout. Perhaps 1V from the meter?
This means your basically wasting 4V of your batteries power. So, 3.6V + 3.6V - 4V = 3.2V which is probably less than the voltage required for a 1.5A drive level.
To confirm this, try using 3x18650 and i'm guessing you will get the 2.2-2.3A you saw when close to short circuit. Which... btw is too much for a M140 i'm guessing. Use a resistance such that your short circuit current is ~1.5A.
What you need is a low dropout driver as well as a low voltage drop method of measuring current. In addition, you need to make sure your 2x18650 are fully charged such that their voltage is at max.
I'd perform the same test again with the exact same setup (with the m140, 2 x18650 + multimeter in series) and use a second multimeter to measure the supply voltage of your 18650s under load as well as the voltage as the voltage drop across the diode vs. what you should expect at 1.5A. That should give you an idea of where you are at.