OK, my initial analysis for the clamps in ice water was flawed because I was comparing convection through the air versus conduction through the water. But here is what I am trying to say:
First, the output data of your experiment is the temperature rise. Everything should be expressed in terms of change in temperature. So if you compare a change in temperature of 2 degrees F to a change of 40F, say, that's a 5% effect. That computation is independent of the temperature scale.
Second, as Golddigger suggested a long time ago, what you would like to measure is the temperature rise at the middle of your conductor when there is no axial heat flow. He suggested the typical way to do that is to use heaters on the boundary of the test area to keep the boundary at the same temperature as the test probe, so that there will be no temperature difference between the two, and no heat flow.
That sounds complicated, so I'm suggesting that with your clever idea of using ice water you can instead fix the boundary temperature at two different values and measure the respective temperatures at the test probe. Then you can extrapolate what the temperature at the test probe would be if there were no axial heat flow to get a heat-sink corrected temperature rise.
For example, suppose ambient is a constant 62F and you place the clamps in a large body of water at ambient. Hopefully large enough that its temperature will remain constant during the experiment. Then for a particular set of test conditions you measure a conductor temperature of 90F. Now you repeat the experiment with the clamps in ice water at 32F. The temperature at the conductor becomes 88F. So lowering the boundary temperature 30F changed the conductor temperature 2F.
From this we can extrapolate that raising the boundary temperature from ambient to 92F would raise the conductor temperature by 2F from 90F to 92F. Since the conductor temperatures and boundary temperatures would then be the same, we conclude there would be no axial heat flow in this situation. So we use 92F to calculate the heat-sink corrected temperature rise.
I picked the numbers so the math would be easy, but in general it's just a little algebra for the extrapolation.
Cheers, Wayne