If you want to know the voltage across the switch at a known, fixed current, in your case, two different currents, then you have to have two current sources.

One way to build a simple current source is to consider what happens when you put a resistor to common (supply GND) in the emitter of an NPN transistor, the collector of which is connected to a positive supply. You'll see that the voltage at the emitter of the NPN is always Vbe below the base. From E=IR you can determine what the current in the emitter is based on the base voltage, since it's always Vbe below the base.

If you choose an appropriate voltage such that your power supply can provide the necessary current without "drooping," and you choose two resistors, one for the upper current, and one for the lower, and don't forget that they have to be capable of dissipating the power, then you can wire them into a switch, SPDT with center-off, with the common terminal to supply common, through the relay, the two resistors both connected to the emitter of the NPN, and the base of the NPN driven from a pot-controlled voltage regulator supplied with the same supply as your NPN's collector. Put a resistor in the base of the NPN to protect the base-emitter junction, and adjust the regulator such that the appropriate current flows through each resistor, when there's a direct connection across the terminals where the relay belongs, irrespective of the switch position. Now you can measure the voltage in the resistor, or the voltage across the relay, and either one will allow you to compute the contact resistance.

BTW, knowing the approximate contact resistance ahead of time may make it possible to choose resistors that allow you to make easy measurements. The resistance of the relay should be relatively small, though. If the emitter resistor is small, too, then the ratio will be easy to measure. If it's not, you may have to amplify the voltage difference between relay and short-circuit, which can introduce more error. If you think about it a while, you'll see what's important.

RE