I'm sorry, but you seem to be getting stuck on the part that isn't usually the most complicated. Let's see if we can get you past that.
My sims are vacationing in Dragon Valley, I did this up real quick so the screenshots are not going to be very pretty. They are staying, as offered by the Traveler mod, in some cheap not very nice looking house. I want them to purchase a vacation home and move into it so they can then make DV their permanent homeworld.
I have changed the lot type on the home I want them to purchase to Player Ownable. They can purchase it from the Real Estate and Travel Services menu by smartphone.

Once they do so, they are still living in the cheap rented house but the map tag for their new vacation home looks like this with the star icon that wasn't there before.

Back on the
smartphone, this is now what the Real Estate and Travel Services menu looks like. I will select Move To.

I am presented with a map showing all of the possible properties they can move to in that world -- there is only one, the vacation home they just purchased.

I am then asked for confirmation that this is what they want to do.

Once I confirm, the move animations start and my sims begin making their way over to their new vacation home to stay thus leaving behind the cheap not so nice one forever. In Map View (not Edit Town), I can see my sims are still just outside of the cheap rental about to make their way to their new vacation house. The vacation house they are moving into shows up with a regular "this is your home" type of map tag.

Once they actually move into that vacation home and the animations finish, this is where I would expect Traveler's Change Hometown to work, if it's going to work with the two worlds that are in play at all.