Can eddy current sensors measure through a thin metallic sheet?

It's all about the size of your target, and signal to noise ratio.

Your eddy current sensor will detect two returns, the 100um of copper foil, and your real target behind it.

The copper will return some signal, which will be your 'noise' level. How constant it will be depends on the constancy of distance to the sensor, and temperature of the copper, which affects conductivity. Why worry about how constant it is?

Against that background, you will need to detect your real target. The outgoing 'illumination' will be attenuated gong out through the copper foil. The return signal will be attenuated again. As the distance to the target increases, the return signal will weaken further. This has to be contrasted with, and detected over, the noise signal, and changes in the noise signal.

This will be easier if your target is big, say thick copper, than small, say a piece of stainless steel wire. You will only be able to be confident on changes in your target signal that are larger than changes in your noise signal. Given that your noise may be orders of magnitude greater than your signal, this is a tough ask. You will need to experiment with your target and detector, and see whether you can achieve any practical distance at all.

I worked for a while in the food metal detection industry. There, signal to noise ratio, and stability of the noise, was paramount. An example of noise was the return signal from the conductive electrolytes in the mayonnaise in coleslaw salad. After the manufacturer had mixed a batch of one tonne of coleslaw, the ratio of mayo to the non-conductive carrot and cabbage might be different between the first tubs filled and the later tubs . The machine had to detect whether there were any fragments of stainless steel sieve wires (which generates a notoriously weak signal) left in the product, against the changing background of varying mayo concentration. In this case, the relative phases of the return signals helped to discriminate them, and a learning algorithm could keep up with the relatively slow rate of change of mayo concentration.

A 100 micron copper sheet, at 10,000Hz, should be well penetrated by the eddy current. Takes 150 nanoseconds for a pulse_edge to penetrate standard thickness (35 micron) copper foil (I measured this, and Jackson's E&M book agrees).

Thus 100 micron foil (3X thicker) will be 9X slower.

[ Note all this is just electric fields that, given enough time, do move into and through the sea_of_electrons. I offer this, as an alternative to viewing eddy currents as magic.]

Now you must determine how the "metallic object" will affect the stimulus energy.

Perhaps you need a drive coil and a Sense coil.