Years ago I learned an important lesson on that regard
@Skynet. Because of the increasing quality of the mobile phone cameras, I left a mini camera unatended for long. Unlike Li-ion/Li-po batteries that tend to keep charge for longer, Ni-MH batteries tend to drop charge,
and capacity after they fully discharge, in shorter time when stocked, and so we have to keep that in mind. When I got back to my AA camera batteries they were discharged. I charged them and checked voltage, they had about 1.6 v each, which was fine, yet the camera refused to work. I tried all I could (the charger you recommend above would have been very useful) and was about to throw the camera in the bin, when I found out from reading around that a multimeter is not capable to determine the
capacity in the batteries. The voltage can be correct but if the battery can't hold the minimum required capacity then it is time to dump it.
The above is good for any device and any rechargeable batteries. So, if your Backlit remote is not holding the batteries long enough it might well be because of the batteries capacity, even when you are sure they get to fully charge to their maximum voltage, and you then need something more technologically advanced then a multimeter or a common simple charger to check that out.
Because of laptop cells I had to do a recent research on the popular 18650 cells (did you know a Tesla car runs on over 7000 of these tiny things wheighting around 540kg?). And I found out we should stay away of some cheap sold cells advertised to be ultra high capacity, such as 9000 mAh or over, because when tested they can be in reality less than 2000. So unfortunately the fact you bought new cells for your remote is not a gurantee that you get what is mentioned in the store package. Before dumping your remote it might be worth it to proceed to further battery testing.