You don't have the budget for pro grade equipment to do this job (e.g. a Keithly SMU would be an order of magnitude more expensive) so will almost certainly need to DIY, as hobby grade battery discharge testers capable of detailed logging are typically intended for far higher capacity batteries, so cant provide a suitable load.
Look at the battery datasheets! If you want to reject out-of-spec cells, you need to test with the manufacturer's specified load, not your device's actual load current. However if your device exceeds the highest pulsed load in the datasheets of good quality branded cells, you have a problem!
Most if not all the datasheets seem to show data for resistive loads rather than constant current loads, and the lowest resistance pulsed load I have seen is 100 ohms for an Energiser CR2032.
I agree with Fungus: use an Arduino or similar. The higher resistance continuous loads could be directly switched by MCU I/O pins sinking their current, but anything under 1K is going to need a small MOSFET, with its gate driven by the I/O pin. If the load resistance is accurately known, the discharge current can be calculated from the terminal voltage, and an AVR Arduino's ADC would be good enough, if buffered by a unit gain OPAMP so its sampling current doesn't load the cell under test.
If you want a standalone unit consider hacking a M328 style component tester kit. Fit your discharge resisters in place of the kit's resistors between the MCU and the test pins, hack in an OPAMP buffer in one of the ADC inputs from the test pins, and hack the other two to drive MOSFETs for the low ohm pulsed loads, link all the test pins together and connect the cell holder between there and ground. For firmware development purposes it can be treated as an Arduino with no bootloader.