If you're going to haul it to the gate for checking anyways, you can ask your gate attendant if there are any empty seats on the plane. If there are, I believe they will allow you to use them for your child's seat. If not, then you can gate check it (although again, I wouldn't recommend it, because it may cause the seat not to be crashworthy after that point. Gate checking removes some of the risk, but the seat can certainly still be mishandled.)
Also, safety advocates recommend using a seat on the plane if at all possible -- it's not so much the drop from ten thousand feet to worry about (as that will be fatal for everyone,) but rather the runway emergencies and especially turbulence, which can easily rip a child from their parents' arms and send them tumbling into the ceiling or several rows away. I'd never thought twice about lap children before getting a major carseat education and then becoming a CPST (tech,) but now the thought of having a babe in arms makes me shudder.
It's totally your choice to make, but it might be worth relaying these items to your wife so that she can make an educated decision.
Depending on your seat, it may fit through the scanner or it may need to be hand-wanded, so leave yourself a bit of extra time. You can buy a luggage cart, a device to attach it to your rolling carryon suitcase (and put your kiddo in it for use as a stroller!), or even bungee it to your stroller to haul it through the airport.