Instrumentation Amplifier output mysteriously increases upon connecting across a resistor, compared to connecting the same voltage directly across it
This is caused by the common mode voltage at your input.

A real differential amplifier has some sensitivity to the common mode voltage at the input that results in a signal at the output even though there is no differential input voltage.

It can be caused by mismatch of the resistors or the result of the offset voltage of the opamp changing with the common mode voltage.

Since you are doing a simulation with "perfect" resistors it is presumably caused by the opamp.

You need to research CMRR (Common Mode Rejection Ratio - Wikipedia) - it will be specified on all opamp data sheets.

In the real world you wouldn't be able to use such a small signal as the 1uv that you are using because of such imperfections. The resistors would not be perfect either and would also contribute to the CMRR.

If you rearranged your circuit such that the resistor under test is floating (i.e. not connected to ground) and the input to the amplifier was connected to ground there would be no common mode variation and you would not have the problem.