I think I see what you want to do. You want an actual 4-terminal meter whereas this one is bassically a two terminal device.
I think you want to do the following. Delete the horizontal wire going to pin #1(+IN). Take (+IN) and this will be what I would call the V+ terminal for the DUT. (I+) will be the top left terminal of the DUT as shown.
Move the ground symbol about 1/2" to the right on the schematic that is near the Vout label. Remove the resulting horizontal line. So that 2, 3 and 4 becomes -IN.
At the DUT. Relabel the left ground as I-. At the DUT V+ and I+ are connected together and I- and V- are connected together.
As an aside, 10 mA of current is sometimes not enough wetting current. It would be more advisable to measure the contact resistance at the currents encountered. For reverence, you would not use a 10 A relay to switch a 1 mA current.
See if there is a minimum current suggested for the pins of the connector you are using.
The output reference ground would probably be better taken from the new pins 2,3, and 4.
I hate drawing. It's like work.