May be too late already, but my approach we be...
Skip the collapsible luggage cart and replace with stroller. Don't check a stroller at the ticket counter (assuming this was your plan, since you didn't mention a stroller otherwise). Rather, put the carseat and diaper bag on the stroller (assuming it's a carseat-stroller system; if not, the snap'n'go is a good affordable option, as mentioned). Wear the backpack and roll the stroller and suitcase. If you check anything at all, I would suggest checking the suitcase and not your stroller.