Electronic – How to check with a simple multimeter whether an old lead-acid battery and its charger are still working

Your dumb charger is just providing a slightly regulated voltage. It works due to the combined resistances of the charger and battery limiting the current. As the battery voltage gets close to the charger, the voltage potential is less. So with a fixed resistance, the current drops.

I doubt that you can modify your current charger to push 38 A up to 14.4 V. This could only happen if the combined resistances and voltage difference allow sufficient current flow.

Most likely you will need a higher voltage source and then circuitry to make a constant current source to provide a fixed amperage. Shutting it off can be done with a simple comparator circuit. Put a 10k trimmer pot across the battery voltage and run it into a comparator along with voltage from a zener regulator. Then you just adjust the voltage divider from the pot till it trips at the correct battery voltage. Use that to turn off a high current MOSFET or something similar. (Might use the same MOSFET to control the constant current.)

It sounds like both batteries are likely end of life, but testing them is a great way to determine if there is useable life remaining. I'm assuming that you don't consider the burglar alarm to be mission-critical, as using a marginal battery in applications that really need reliable power is a bad idea.

First, measure the terminal voltage during charging (but after they've been on the charger for at least 24 hours). The terminal voltage should be between 6.0V and 7.5V. If it's less than 6.0V, you have a shorted cell and should recycle the battery; it's done.

Next remove the cell from the charger and measure the terminal voltage. If it is 6.0V or greater, there may be some useful life remaining in the battery. Reference a typical VRLA battery terminal voltage vs SOC curve below (note that this is for a healthy battery).

Finally, connect a load such as a 6V light bulb across the terminals while measuring the terminal voltage. If the battery lights the lamp and the terminal voltage stays above 5.8V or so (assuming a ~500mA load), you're probably OK given the caveats about not using marginal batteries in important applications.