Using non-magnetic tweezers is not a bad idea at all, ardly. Thank you for the suggestion. I googled a bit and there are aluminium and copper tweezers around, but not so many and mostly relatively expansive. However, I already ordered hall sensors just in case and I'll keep that option at the back of my mind.
Capacitive sensing did not really work out in the end, unfortunately. I experimented with different electrode/capacitor geometries, resistance values and different samples/organs. I was surprised how well this works with big/medium sized items like fingers or scissors or so and this might come in handy in one of my next projects. But it looks as if my "organs" were just too small for that kind of sensor.
I also tried that stripe geometry, that wvmarle suggested. I used a standard stripboard for that. I did not get it working, but maybe I did not get exactly how this was meant. Could you please clarify this, wvmarle? Should the stripes be alternately connected to the measurement electrode and to ground? Or somehow different?
Right now I think about going back to the original plan and short-circuit some wires by placing the "organs" on top of them. I will never get 100% reliability there, but maybe there is no easy way of achieving that and I can live with it. I'll experiment a little bit more with it different geometries.
However, I still keep all the other suggestions in mind.