Constant Current Source using Fixed Voltage Regulator
Depending on the regulator you can indeed also use a fixed regulator as a current source. The only disadvantage is that the voltage drop across the shunt resistor (R2 in your schematic) will be higher.

That LM1086 looks suitable to me but it will drop 3.3 V (the regulator's set voltage) across R2. Variable regulators typically will output a smaller voltage when the ADJ pin is grounded (like 1.25 V for the LM317) and will therefore also drop less voltage across R2.

So depending on how much voltage the laser needs, you might have to increase the 10 V supply voltage. For a 10 V supply I guess if the laser's voltage is less than 10 V - 1.3 (dropout voltage) - 3.3 V = 5.4 V this should work. R2 would need to be 3.3V/1A = 3.3 Ohms, not that it will dissipate 3.3 W so use a 5 W resistor!!!. Also the LM1086 will need a proper heatsink.