I'm sure there are a lot of "experts" out there that will want to comment to this because they surely know better, but. . .
Your present converter is fully capable of charging your lithium battery. A lithium battery has a "happy place" for high to low operating voltage. The internal battery management system (BMS) inside the battery will manage this. Your charger will operate in this zone, although not as efficiently as a charger designed for lithium. The charge curve is a little different and voltages typically a little higher initially with a lithium designed charger. Short term you are OK, but I do suggest long term, you upgrade to a lithium capable converter.
All this being said, you should not be experiencing significant "depleting" as you put it. You may have a different issue. Did you have batteries dying prematurely with your (original) lead acid battery?
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