Is that posible with an arduino?
Yes you need to apply a known voltage to it and measure the current. Normally this would be done using a small resistor and measure the voltage across it. However as you coil is already in the 1R region the currents involved would be huge. As well as only supplying 5V and arduino output can only source 20mA of current and this is more than most micro controllers. You would be much better off trying to encode some sort of key so the controller knows what coil is used, or a coil code to enter.
is there a booster board that can maybe be installed along side that might give 2.5 volt boost to whatever voltage is applied?
Anything you do you will have to design yourself. It looks like you need to supply a current of 2A with a variable voltage of 3 to 8V. Such things are available commercially but cost 100s of $. It is not a simple thing to design.
Your problem is not just voltage it is current capability as well.