Thanks all for your help. @WBahn, that should do it.

 

Reasonable concern. Put a suitable switch in parallel with the meter and close it before powering up the circuit. Then after the wire is at temp, open the switch and make your measurement.

To keep the rapidly cooling wire at operating temperature a little longer while you measure using this method, sandwich the wire in some fiberglass insulation and use the min/max feature on your meter to capture the data.

 

Well...

If it was an "ideal voltmeter" with infinite input impedance, you would be measuring the drop across an open circuit and you would measure the power supply voltage.

But in reality, the meter will have something like a 1Mohm input inpedance so you will form a voltage divider with the series resistor. A simple voltage divider calculation will give you the answer.

 

Add sufficient known resistance of proper wattage in series with Rx to give proper cutting temp. like a dull red. Measure both resistor voltages, calculate current in known R, Then calculate Rx.