My wife's first deer rifle was a Remington 660. It had an 18 or 18.5 inch barrel. It was pretty loud, and I suspect she was only getting around 2600fps with 100gr bullets out of it. But she put a lot of venison on the table with that little rifle.
On the other hand, you asked, "is there a better caliber for this in the .308 family?" In MY opinion it's the .308 Winchester itself. It just seems to me that when you're talking about a "light, short deer rifle" you're not talking about a 400 yard hunting rifle, you're talking about a handy, easy to carry, quick pointing rifle for use in the timber (sometimes called a "brush gun"). For those purposes, I'd prefer a rifle that throws a heavier, larger bullet than what you can throw from a .243.
But that's just MY opinion. You have to understand I have a real affinity for 30 caliber cartridges. My first big game rifle was a .308 Winchester (model 100 Winchester with a
20" barrel by the way) and my retirement rifle is a .308 Norma Magnum with a 25" barrel. I've had a few smaller, and larger caliber big game rifles over the years, but I always come back to the 30.