The exact same thing is going on with my daughter, who is also 4.5.
We have discovered that constipation and the big colon that quickly results is really common in kids 4-7. It is not psychological but a physiological development issue, that begins when they avoid pooping because it hurts or they are too involved in something to bother to go to the bathroom.
For us, it started in April and we have been managing it with Miralax (or the Miralax has been managing us) since then. Another thing we've discovered and confirmed w/Internet research is that this type of constipation (w/enlarged colon) usually takes a long time to resolve.
When we started w/the Miralax, the Dr prescribed 1 cap/day. After that the poop seemed to be leaking out just as you describe so I stopped the Miralax, convinced it had given her diarrhea. We too were on some level convinced that she was just too lazy to go, but now know that's not the case. We went to a different dr. who took the time to explain what was happening in her colon. That amount of Miralax (1 cap for 2 days) loosened things just enough that poop seeped around the blockage, and out, uncontrollably. The blockage remained. Your daughter really can't control the seepage.
The solution for us was yet MORE Miralax (and I hate medicines...) but that did the trick. With one cap a day the blockage passed after about a week, and then when it happened again about a month later we went up to 1.5 caps for a few days...which cleared the second blockage. The colon must stay clear, and that's the only way it can heal back to its normal size. We still give her 3/4 cap to 1 cap per day (on alternate days), and the plan is to slowly decrease the dosage over time...like months of time.
From personal experience and Internet research I can state that Miralax has tricky dosage!! But it seems that your daughter is not getting enough to clear the blockage. The good news is, once the blockage is gone the constant flow of poop is gone too and life gets a lot easier.
I'd be happy to share my practitioner's name if you'd like to consult w/her. She has a lot of experience with children's health and really has made us feel like we are in capable hands. Her office is in Albany but she will consult over the phone too. Also, you can call me if you want to discuss poop in detail (I know)...just email and I'll email back my phone number.
Let's keep in touch if you like to see how things go! I know this has been one of the hardest challenges of my parenting life. It's frustrating and seems endless. Thanks for posting.
J.