Backup Shattered iPhone 7 Plus

We just adopted a puppy who seems to have it out for my husband. He got ahold of his iPhone and pretty much chewed it completely on all 4 corners. It was in a case but it's the screen is absolutely chewed up including the home button. We have insurance so we're getting a new phone but the issue is that he hasn't backed up it up in a few months. When we plug it in to my laptop/iTunes it's apparently not registering my Macbook because it wants me to touch "trust" this computer on the phone. Obviously I cannot do this. So the good news is that he didn't ruin anything inside the phone. I've been researching if there's any way to get a backup of his phone but everything I've found gives how to do this using a bluetooth keyboard but you have to be able to activate VoiceOver/Siri which neither are available to me. How do I get this backed up? Is fixing the phone my only option because I have to return the phone once I get the new one so I really hate to spend the money. I feel like there has to be a way to do this but I'm not finding anything positive online........