To capture the rare insects that appear on palm trees, such as the lamprimaeus lucarne or the giraffe lucane, the most effective is to go to a mysterious island with ACNH Nook Miles tickets. Whether it's catching those missing from your museum or stocking up on Animal Crossing bells, here are the steps to follow:
Wait till after 7:00. From this time on, butterflies and other grasshoppers no longer appear, so lucans and beetles are more likely to show up. Note that some bugs (the beetle and the kite lucarne) are only there from 11pm, so you will have to wait for that time if you want them too.
Bring in your pockets a few palm trees, an axe, a shovel and a net (or several!).
Go to a mysterious island and make sure there are insects on it (some islands, like the bells, don't have them). The best choice is a bamboo island, followed by an island where there are only trees, but all the others can do the trick.
Pick all the flowers, cut down all the trees (unless you want to catch insects on them) and remove the trunks, to prevent insects from appearing in these areas. You can also place objects on the rocks so as not to leave room for the ligies. All this will force the insects to show up on the palm trees.
Plant the palm trees you have brought in so that you can see and catch the insects that will perch there.
Beware, rare beetles are very fearful! Walk slowly (don't push the joystick to the bottom) and be on the lookout.
When you see an insect to catch, prepare your net by pressing A and approach slowly. Stop when the insect comes to a stop, and resume your approach when it starts to swarm again. This will prevent him from running away.
Continue to tour the island, catching or scary the few insects that roam the ground and mosquitoes.