Nah, the switch lite doesn't dock to a tv and in fact it's pretty much impossible to get it to work with a dock at all. it's missing the internal components necessary for it to connect to a dock and even be able to output to the tv at all.
From my understanding, there is a chip in the dock itself that "received" the video input signal from the handheld, but both the dock and the switch lite console need these chips to be able to communicate and feed the video/sound through to the tv. I could try to explain that here, but I'll do one better and copy paste this reddit comment I found where a knowledgeable user explains this better than I ever could:
So yeah, even if you could somehow open up the thing and get your soldering iron out and get it setup, it's a lot of work and know how that I'd say isn't worth risking breaking your system. So to answer your question about a switch lite dock hack or switch lite dock mod, MAYBE, but not worth it. Your best bet/cheapest option is probably to buy a mini projector to connect to your phone and set your phone up to record your switch lite screen. Then you can project the image onto the wall or something. Not the best setup, but if you REALLY want to do something, that's an option.
Best of luck.
