Can current still be (alternatively) measured with a blown multimeter fuse?
Yes, provided that:

1) The fuse blown only affects the current measuring function and not the voltage (and if needed, resistance) measuring circuits of the meter.

2) You have a small value resistor of sufficient power rating which you can insert into the circuit to be measured, and either know it's value by specification or can measure it with desired accuracy using the resistance function.

3) The circuit will still function, and in a representative way, when subject to the voltage drop produced by the instantaneous current through the sense resistor inserted to facilitate the measurement.

If all these conditions are true, the current is the voltage drop measured across the resistor divided by the value of the resistor.

Appropriate sizing of the resistor will depend on the current you expect to measure and the tolerance of the circuit under test to this current-dependent voltage drop.