0-5V Voltage controlled Constant Current Source with High & Low current Offset
It seeems like you have two issues to solve. Firstly you need to convert the linear range of 0 to 5V to currents ranging from \$I_{THRESHOLD}\$ to Imax (2.5A). This is done with a summing op-amp circuit and a further inverting op-amp circuit to remove the signal inversion in summing op-amps. There are other op-amp circuits that are simpler in construction but this is the easiest to visualize: -



The output from the above circuit is -(5mV + 2mV) so you can apply your control signal to one input and have your offset voltage set on the other input. Remember this is inverting so it needs a further inverting op-amp to return all the voltages to positive values.

Next, you need a voltage to current converter: -



This works so simply - quite literally I (current into collector) is Vin/R. Ignore the two resistors that drop the input 5V down to 1V - this is the closest circuit I could find on the web. Vin is shown as "5V" on the picture.

You connect the laser diode cathode to the collector with the anode up to the positive supply. You choose the resistor to suit the input voltage (say) 5V to give you 2.5 amps in the laser so, R = 5V/2.5 amps = 2 ohm.