Electrical – Turning an unneeded outlet into a light switch to power LED lights

I'm just thinking out loud here. It is hard to state anything with certainty in a situation like this where there is no guarantee of what somebody else did with the wiring a long time ago...so forgive me if this suggestion isn't even worth what you paid for it. ;)

From what you have said here, it sounds to me like that red wire makes its way back to the lights, and acts as the "hot" for the lights. Who knows where the lights are getting their neutral side connection. Doesn't really matter. Obviously it is getting there somehow.

So to make the lights light, your existing switch is getting "hot" from one of the two black wires, and is supplying that to the red wire.

So think of your red wire as the "hot out", and ONE OF the two black wires is your "hot in".

So you need to use a meter to see which one of the two black wires is your "hot in", and treat it that way and cap off the other black. Then use your red wire as the "hot out".

Again, this is just me thinking out loud about what I would try. I take no responsibility for any bad stuff that might result from it. All possible disclaimers apply. ;)

Your light and switch arrangement needs hot and neutral to function. You tapped it off the light switch in the bedroom, which only has hot and switched-hot (marked as white with black electrical tape). That is not a good place to get power. Here's what's happening:

With the bedroom switch on, the two pins on the switch are shorted out, so both power wires to the closet have the same voltage. It can't function.

With the bedroom switch off, current flows from hot, through your closet switch, through your closet lamp, and then through the bedroom lamps and to neutral. The lights are in series. That would light - dimly - with incandescents, but smart bulbs won't like it at all.

You cannot tap power at the switch if there's no neutral there. You should tap it at the bedroom light, where neutral is available.