DS Lite Charge LED lit when battery is inserted, charger is not inserted

This is only a guess, but are you sure you installed the replacement part the right way? To me this sounds like something in the circuit is going the wrong way and the battery is getting connected to the charging regulator instead of the main power lines.

 

The only other thing I can suggest is checking for faulty power ICs, but those are not easy to replace. You could just replace the entire motherboard if you mainly care about the good condition of the outer shell and the displays.

 

I don't think 3rd party DS motherboards exist. The only thing to be concerned about is that the boards might have come from faulty units and been repaired, so you don't know how long it will last, maybe there is some unseen damage. But it's probably not likely, component level repair for the purposes of resale are just not worth it, they probably just buy lots of consoles with broken screens and resell all the usable parts, throwing away anything that's faulty.

 

Have you checked the little water indicator sticker on the motherboard? and the one on the case?