Trying to Replace 3-Way Light Switch
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.  ;)