I need to measure the current in different brands of alkaline batteries - 1.5v AA and AAA, as well as 9v (6F22) - using a multimeter, and a resistor as load. How much resistance must the resistor have in order not to "blow up" both the batteries and the multimeter?
A 4 ohm 10W resistor is good place to start.
Note: Battery Internal resistance has significant influence in battery current.
You also could find Questions regarding CR-P2 battery specification. There are few links including the one referenced in the comments that you might find interesting
You want to measure ESR and also Voltage for State of Charge. There's many ways to do this.
A quick short circuit on the 10A scale might be hard to get a steady reading. It should peak then decay slowly but only be done for < 1 second to not waste power.
The ESR=Voc/Isc for open circuit voltage and short circuit current. Meanwhile it is heating up with Voc*Isc = P power . Alkaline are pretty safe doing this test but also a pretty fast way to kill a battery. But a 1/4 to 1 second is no problem. Even 10 seconds but overkill to get a reading.
What you are looking for is to determine your pyrotechnic resistance ranges and use those using a non-inductive bifilar wound length of wire to give the equivalent resistance as your squid. Then see what the voltage drop ( and thus drop in power P=V^2/R). Now the squid rises quickly in resistance when it heats up so your expected load current should drop until it fires. IF you can measure that as well, then you can now define your power vs time spec.
Since 9V batteries are made from six series 1.5V cells , each cell ESR adds up and a 9V Alkaline cell ought to be able to pulse > 1 A into a short circuit briefly but not 1A at 8V which requires a battery ESR (resistance) almost 10 times lower like that of the Lithium primary cell.
To measure it safely into a meter, choose low resistor values that generate 100mV max and then use a DMM with a 199mV scale or so. Although 50mV Current shunts are pretty standard for low power dissipation, you also need a more sensitive meter or just an old Analog meter with the right Ohms/Volt sensitivity.
Thus with 199mV/2A = 0.1 Ohm that will dissipate 199mV*2A = 400mW. Any resistor that you choose, will operate at 150'C if you sustain the power, but a brief pulse is all you need.
They use thermal sensitive inks and a heater resistor to measure battery power on some which barely get warm so unlikely to generate more than 1/4W but is valid for indicating Voltage for State of Charge(SOC)