Are one-mile trips draining our battery?

Those short trips are not helping any of the vehicle's systems really. Not only is it not charging the battery properly, but the engine never reaches operating temperature and as such any condensation that forms is not boiled off. Over time that can cause rust and internal damage to the engine. My suggestion would be to take the car for a longer drive at least once a week, at least 100 miles, which should be enough to circulate everything and get the engine up to full temperature to enable it to a minimum remove some of the contaminants that build up from the short drives. To go with that I would look at the severe service table for your maintenance because the short trips and hot weather both put you in that category.
Here they are from the service information -

Follow the Severe Service Schedule if your driving habits frequently include one or more of the following driving conditions:
- Repeated short trips of less than 5 miles (8 km).
- Repeated short trips of less than 10 miles (16 km) with outside temperatures remaining below freezing.
- Operating in hot weather in stop-and-go "rush hour" traffic.
- Extensive idling and/or low speed driving for long distances, such as police, taxi or door-to-door delivery use.
- Driving in dusty conditions.
- Driving on rough, muddy, or salt spread roads.

Those should be listed in the owner's manual as well. As for the battery itself, the short trips will shorten its life in two ways. First the vehicle starting draws more current than the 1-mile trip will replace so it will constantly be lower, until you take the longer drive which should recharge it. However while it does that the alternator also looks at the discharged battery and tries to charge it back up as fast as possible as part of the fuel saving strategy, basically it raises the voltage higher and generates more current in an effort to recharge it quickly, then it drops back to a level that equals the drain the vehicle places on it the rest of the time. The constant cycling of high output, then high input with no idle time will actually wear the battery and alternator more.

With a synthetic you could go to the 5,000 miles change for severe service without a problem. If you are using bulk "conventional" oil 3,000 would be okay, but if it is bulk oil at Walmart, it may be sold as "conventional" oil, however it is actually a synthetic blend. It's hard to get a real conventional oil these days in bulk unless it's a specific application lubricant. And paradoxically the conventional oils are more expensive than the blends or full synthetics to boot! I picked up a case of conventional for my garden tractor a while back and for a synthetic 10W-30 it was about $4.00 a quart and the conventional was $8.45 a quart!
Personally, I have always thought that changing the oil and filter more often than "required" was still cheaper than replacing the engine. For instance, the oil change light in my wife's car turns on at 10,000 miles! Now for Chrysler that works fine, it gets the car out of warranty, and they can claim a lower cost of ownership. However, the engine in it is known for having issues directly related to lubrication problems. As such It gets new oil and filter every 5,000 if not sooner. A replacement engine would run me about $4,500.00 at my costs or I can try to keep it alive longer with better oil.

They don't specify a time limit in the service info other than the oil should be changed at least once a year. $4,500 is my cost on a remanufactured engine. No mark up or labor, my wife doesn't like it when I try to charge her anything over my costs. LOL
I can get an unknown used engine for under $200.00 but I'd rather not use them.