Hey! I'm not sure what the guidelines are for Delta, however, when I worked for United Airlines, I noticed that the soft sided ones were best because, they could be collapsable (if needed to fold and pack into a suitcase, etc.), were lighter to carry (airport terminals can be long and tiresome to walk through), and can be smooshed a little (if needed) to adjust to fit under the seat in front of you.
Basically, a small duffle bag would fit under the seat in front of you, so would a small, soft sided pet carrier. I would get the smallest soft sided carrier, with enough room for your pooch to turn around and stretch a little.
Also, try calling Delta again, and hopefully you will get a better agent to talk to. If you are super worried about it, go into your local Delta Ticket office, or the airport ticket counter, BEFORE your trip and have them measure the carrier you choose.
~Debbie
