If you own a Crystalline Aura active, you can also access your personal storage with your pet. All you have to do is open the pet features that grants the same access to your storage as speaking to a Storage Keeper. The only disadvantage is that you'll need an Crystalline Aura active, which is a premium item.
You must ensure that you have access to your personal storage prior setting out on an adventure to ensure that you don't have a jammed inventory.