Hey there @Foulcat, welcome to the site! To export items from your adventure inventory, you'll have to take your adventuring pet back to the Entrance Gate (at Gateway City), and click "Finish Adventure." Then you open your adventure inventory, and you'll be able to select items and send them to your main inventory.

If you found a Blue Pixpet Bed in the adventure, you may want to hang onto it since there's an NPC who will ask for it, but it's up to you.

Please let me know if anything I said was confusing or if you have further questions! ^^