Is there any repeater with power amplifier?
Without getting into a lot of fancy analog conditioning, I think that the solution I'm proposing should work for you.  But you are going to have to add one more wire to your cable.

You have two problems,

1) The voltage drop in the power conductors is too high and the final voltage at your device is then too low for proper operation.

2) The voltage drop in the Sensor Ground wire is going to add an offset to your analog signals.  This degrades your accuracy - introducing possibly significant error to your measurements.

My approach would be to install a 5V linear regulator at your device.  You now feed perhaps +12V down the power wire and it is dropped to your desired voltage right at your sensor.

The power supply that feeds that regulator is floating.  That is: it does NOT share a ground with your system until it connects to the sensor - right at the sensor.  This is the extra wire that I mentioned above.

You now have 5 free wires coming from the sensor: the signal ground wire plus 4 analog signals.







simulate this circuit &ndash; Schematic created using CircuitLab

I've shown a generic voltage regulator block.  You'll need to add the appropriate bypass capacitors as well as choose the most suitable part - the 7805 may not be the best part for your purposes.

There will be a voltage potential difference between your signal ground wire and the negative lead of your power supply.  This voltage difference is why it is important that this power supply ground be connected only at your sensor.

Because there isn't any current flowing on the Signal Ground wire, your analog signals are presented to your measuring device with the maximum precision.