"I am hesitant to ask for 3 free round-trip tickets"

Well, good, because I don't think you're entitled to them. But aside from that... have you ever considered asking a friendly-looking solo traveler to help you out? As a single traveler, I've actually offered assistance to a couple of single moms with two or three kids. I admit that I did it with an ulterior motive: To pre-board myself (it's nice to just sit and read while everyone else is jamming their crap into the overhead bins). WHile sitting at the boarding gate, I'd make small talk with a mom, make fishlips (love fishlips!) to her toddler, then when the time for preboarding comes, I would offer, "Do you want a hand? I could carry the car seat and the diaper bag or something." (I never offer to touch/carry/hold a child -- no need to freak out a cautious parent.)

My offers were met with gratitude and a weary, "Yes, thank you so much."

Now, I would help someone who appeared to need help, but I admit that the possibility of pre-boarding makes me more inclined to be a Good Samaritan. And helping a mom with an infant and a toddler is always a good thing anyway.

I'm just saying, when the airline staff seems unwilling to cut a parent some slack, consider asking another traveler for assistance. The worst they can say is "no." And at best, you can focus on your kids and some nice person can help out with the non-kid stuff.

Just my thoughts.