Laser driver current and temperature management
You probably can't. And it has zero to do with how your circuit works.
First line from datasheet (seriously!!!):

• Optical output power (continuous wave): 10 mW (T case = 25 °C)

Acting as if you knew that and would be willing to stress your device beyond that:
The optical power output of this kind of lasers is, sufficiently above the lasing threshold, a linear function of the current flowing through the diode. See datasheet p.3.
So, to get 10x as much output power, you need to put in 10x as much current. Simple as that. Following said figure from said datasheet:

at ~40°C and ~1.5mW output, you're currently driving the device with 36 mA. With the right curve, you get a forward voltage of about 4.7 V, meaning your device is converting about 4.7 V· 36 mA = 0.17 W into heat.
15 mW is already out of the range of that curve, but let's extrapolate.
The slope of the power/current curve at 40 °C is roughly 0.3 mW/mA. You need around 14 mW of additional power, so that's about 46 mA more current, or (36+46) mA = 82 mA in total. Right chart tells us you'd have a forward voltage of ~ 5.2 V there. That means you're basically converting (nearly) 5.2 V * 82 mA = 0.43 W into heat here; that's 0.16 W more than you're currently doing.
The danger here is that you have a control loop that keeps the output constant. Now, when the temperature slowly rises, you'll need to push through more current to get the same output. That in turn will lead to more heat production, will lead to higher temperature, and will in turn lead to your control pushing through more power. That'll not end well for the diode.
conclusion
By far your best bet with this specific laser diode is to:
Actively cool the device, possibly with liquids, to keep its temperature as low as possible, so that you get the &quot;better&quot; P_out/I_in curve
Realistically, get a stronger laser diode. These might not be fun to hit your eye with.