I would be very worried about leaving a digger out in a pen when I am away from the house.
digging is natural so it will be hard to totally train it. You should absolutely interrupt digging and bring them to where they can dig (the safe spot). but you need to supervise and catch them in the act all the time. And weather you go this route or just "no digging at all" (which would just be interrupt digging and redirect to a safe activity) there is ALWAYS the possibility that the dog digs out of the pen one day.
There is a way to dig-proof a pen but either digging in and putting fence/chicken wire or rocks. But it would be a big job. Or you can use a different ground for the pen.
also totally not fair to your older dog to be penned with a puppy all day. he deserves a quiet safe place of him own. puppies can be super super annoying especially to an older dog. plus the puppy should learn to be "alone" in a crate not always attached to the older dog as part of growing and becoming his own independant happy well rounded dog.
in fact you even posted to say the older dog seems exasperated with the puppy at times and looking for tips on the front - tip #1 never leave them together un-supervised. you need to be there to supervise interactions and step in when needed to redirect the puppy