I agree completely that running Catalina on an elderly Mac with only 4 GB of RAM would likely be an exercise in frustration at best. Personally I would never attempt to run anything other than the Apple apps that "come in the box" with macOS on any 4 GB machine, and then I would expect poor performance. My focus on the DIMMs themselves arises out of the fact the "memory test" portion of the boot process takes place in firmware and once the alarm is raised, the boot process stops. The OS itself can't be loaded and therefore cannot be identified or tested so the Mac has no way of knowing what OS is installed.
From the sequence of events, it occurs to me your DIMMs might have been subject to ESD (Electro-Static Discharge) damage. It has happened to me in the past. All DIMMs are vulnerable and strict anti-static measures should be followed to prevent their being
fried in handling. Testing using your daughter's DIMMs is a good idea, but under the circumstances, I would review
anti-static handling procedures and follow them assiduously to be sure her DIMMs didn't get
fried as well. (Now the questions are: "where did I put that anti-static mat and wristband, and do I still have any anti-static spray?"
)
