Sounds like unnecessary work at this point when you already have a lap child ticket issued for the relevant flights, especially if you’re ok with the price of the lap child ticket, have the lap child ticket number and can validate that the lap-child ticket details are correct and in alignment with the accompanying adult’s booked-and-ticketed flight details. At most, at this point, my recommendation would be to call up Swiss and make sure that they have the lap child noted there with an accompanying ticket number and ask that they look at it to see that lap child booking and lap child ticket is ok.

Sometimes the price that LX wants for the lap child ticket is different than what UA wants to issue the lap child ticket, so I sometimes call or go around to fish for a better price for the lap child tickets. But once the lap-child ticket is issued and I can see it is issued properly to cover the relevant segments for the separately ticketed accompanying adult, I would just leave it as is unless there is a big price discrepancy.

The lap-child ticketing carrier is more likely to mess things up when it’s not the operating carrier for the accompanying adult’s tickets than when the lap-child ticketing carrier is the operating carrier, so make sure to have the lap child ticket number and details and check those are done right to align with the adult’s flight booking.