If APNAP, my fear is that this gives the NAP a chance to choose a different card once they see what I reveal.
It is done in APNAP order, but the other player won't be able to change their choice.
101.4. If multiple players would make choices and/or take actions at the same time, the active player (the player whose turn it is) makes any choices required, then the next player in turn order (usually the player seated to the active player’s left) makes any choices required, followed by the remaining nonactive players in turn order. Then the actions happen simultaneously. This rule is often referred to as the “Active Player, Nonactive Player (APNAP) order” rule.
Example: A card reads “Each player sacrifices a creature.” First, the active player chooses a creature they control. Then each of the nonactive players, in turn order, chooses a creature they control. Then all creatures chosen this way are sacrificed simultaneously.
101.4a If an effect has each player choose a card in a hidden zone, such as their hand or library, those cards may remain face down as they’re chosen. However, each player must clearly indicate which face-down card they are choosing.
101.4b A player knows the choices made by the previous players when making their choice, except as specified in 101.4a.
So in this case, both players would choose a card from their decks in APNAP order and keep those face down. Then you would put those cards on the top of your decks simultaneously, then they would get revealed by Lantern of Insight.