Update: I believe that Archie and I have developed a way of communicating about whether he wants to come out to play or not.
When I see him up and about I put my hand in his cage. He always does one of two things.
1: He goes back in his house. In which case I leave him be and will try again at some point later.
2: He comes over to me and grabs onto my index finger with his front paws (it is so cute I could squeal). He then stays perfectly still to allow me to pick him up.
He comes out every day, but only on his terms lol. I have also discovered that his favourite treat is dried mealworms