Next time, you might want to try to avoid a 3 hour drive by checking to see if there are other authorized service shops in your area signficantly closer to your home.
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Calling an Apple Store on the phone is next to impossible unless you have a direct line to someone at the Store.
But, it is easy to contact an independent authorized service shop in most cases via phone, email or text. The idea here would be to contact the shop in advance to make sure that they have the part that you need in stock....before....you head to the shop. Please note that some shops will not order a part in advance unless you pay in advance.
You haven't provided any details on your phone as far as the model, so we don't know whether you have a recent model or older model. A Store can always be temporarily out of a battery as well, even if it is a popular recent model.
Whether or not I have used the Apple Store or an independent shop for service / repairs, I've always received an email survey from Apple asking about the recent service visit. That would be one way to let Apple know that things fell short on your recent visit.
You don't indicate whether you will be returning to the Apple Store to pick up the phone once the replacement battery has been installed, but if that is the case, I would ask to speak to the Service Manager or Store Manager when you return to the store to pick up the phone.
Otherwise, not much else we can do to help. We are users......just like you.