funny you should post this, I just bought a used Marine VHF handheld radio on ebay, and Im sitting here testing NiCad batteries while I type.
A battery tester will only give you some indication of the present charge left in the battery. As the battery charge is used up the internal resistance does increase (as you mentioned). The result is when you hook the battery into a significant load, the voltage across the terminals is lower than it would be if the battery was fully charged.
For non-rechargeable batteries this is about all you can do. Momentarily connect the battery to a significant load, and measure the voltage. The resistance (load) that is significant for a battery depends on its size (capacity).
For rechargeable batteries, the best way to test them is to cycle them (this is what Im doing right now). You charge the battery fully, then you connect a constant load. For example, I am testing a 7.2V 600mAhr NiCad pack. I have it connected to a 10W 10 ohm resistor. I connect it to a DMM, write down the time. When the battery voltage falls to 6V (about 80%) I stop the test and note the elasped time.
The capacity of the battery is the averege current * the time duration. In my case the 600mAhr battery is putting out about 420mAhrs (lasting about 35 minutes). Not bad for an older battery, but not new either.
If you dont have time to sit and check the DMM every few minutes, there are data logging volt meters (like my Keithley 175) that you can setup to record the voltage once a second, or once a minute, or once every 10 minutes… so you can let the test run and come back later and see how long it took for the battery to drop.
Usually you run the test with the battery loaded at a rate that will discharge it in one hour. For something like the Exide batteries in the FIRST KOP, that would be an 18 Amp load, and the battery should take about an hour to drop 80% of its voltage. (sometimes finding a good load is a challenge. 18A * 12V = 216W. That would be a pretty big resistor (expensive). For large 12V batteries, you can get 12V 50W light bulbs that fit in a regular light socket. They use them in campers and RVs - most RV stores carry them, and some places like Home Depot. Car headlights work well too)
Also, if a battery has not been used much, you might need to cycle it a few times (charge, drain, charge, drain…) to, well sorta wake it up.