I bought an inexpensive chinese one recently. Did the usual check on specs and reviews on amazon and YT. Honestly, it was more a guessing game than a highly-supported choice at the end, and I'm sure the prices and specs change all the time. There are tons of those. Mine works fine with just one semi-annoying quirk. And the current values displayed are fairly accurate, even on the lower end; but the voltage is right on, which matters more for battery charging. In fact, I also have some older much better lab-quality ones but I end up using the new little guy most of the time, way more convenient and especially much smaller on the bench. One thing I would do now, though, is try to find one that has locking adjustments for V and A, cuz you really don't want to bump the voltage knob by accident and have it go to 30V on your lithium battery..., but I suspect those are probably found in a much pricier category.
You can charge a number of types of batteries with just a CC then CV scheme with the typical PSU, and some of the Lithium types are easy in that respect. You basically have a fully-adjustable battery charger.
PM me if you'd like to know my specific model but, honestly, I can't really recommend it much over others.
added: as to your specific situation, keep in mind that there are 2 issues there, one is the recognition by the charger -easy to fix usually and what we're talking about above- but the other one is that some lithium batteries will get permanently damaged if discharged too much and I don't think you can do much about that.