How about this: use an RC circuit to average the current and thus replace the mechanical parts of an analogue multimeter. I presume with careful adjustment of the RC network, the correct value to discharge the capacitor slowly, but not too slowly.
I attach a simple example of such a circuit I just came up with. We use a resistor Rsense to sense the current going into the load. Ideally we would want the Rsense not to impede too much on the circuit downstream, so for example Rsense should be 1/20 or less of the lowest expected Rload.
The very small, floating voltage developed across Rsense is referenced to ground as well as amplified (10 times in this example) by a simple rail-to-rail op-amp. This in turn feeds the RC circuit and in the attached example the voltage over the capacitor in mV is the translated average current in mA (so you could attach a simple voltmeter).