I totally (used to) do this! I discovered it when my DS was about a year old and we moved into a place with a fenced-in front yard. He would only sleep for 45 minutes in the house and would wake up grumpy---I just thought that's how he is! But when he could sleep outside, he'd go for up to 2 hrs 45 minutes sometimes, and always at least two hours. Such a difference in mood.
For us, he's always out of the sun, usually with a screened cover on the buggy, there's a front gate with a noisy latch, and I keep the monitor turned on, on top of the stroller. Like a PP said, I can hear him breathing, when he turns his head, and if the gate DOES open (UPS, mail carrier) it is LOUD!!! and I'm running to the door in a flash. The other thing working against child abductors is that he's securely buckled in, the cover has velcro (noisy) and the brake on the buggy is hard to find and difficult to operate. They'd have to pick the whole darn thing up to get out of there quickly!
My sister has twins, and they took stroller naps for at least a year of their lives, if not longer. She'd bump the stroller up the steps onto her porch and either sit there with them and knit, or go inside and cook (open floorplan, no walls blocking the view). She doesn't have an older child, and it was STILL compelling to get the kids to sleep at the same time. I imagine it is so much more of a priority for you so that you can give DD1 the time she needs with you.
I'd do the same exact thing in your situation. My one other thought is that if CPS ever DID show up, you'd be right there at the door to greet them (probably down the walk before they even got to the door, because you're there watching!)
They'd have nothing to report, really.