I suspect that my charge port is damaged

A temperamental lightning port isn't at all uncommon, so in that respect yours isn't unusual. Although I'm sure there are standard manufactured lightning ports available, it doesn't look to me like Apple used an off-the-shelf one on this iPad. The port itself looks pretty custom, and the replacement procedure is what I would call upper moderate difficulty. I have a somewhat decent soldering station so this is something I would tackle myself, but then again I have taken a professional soldering course so I'm comfortable doing this kind of work. If you're not fairly confident in your soldering ability, I'd suggest finding a professional to do this particular replacement.

On this iPad you've got good news and bad news. The good news is the charging port is actually mounted on a small piece of flex cable rather than the motherboard itself, so it's not quite as difficult to replace. The bad news is instead of using a plug connector, that flex cable is soldered directly to the motherboard.

So in order to fix this, the first thing you have to do is remove the motherboard so you can solder on it. iFixit has a guide showing you how to do that, so you've got a head start with that.

iPad Air 3 Logic Board Replacement - iFixit Repair Guide

After you have the motherboard free, you then have to unsolder the flex cable, clean it up and re-tin the pads on the circuit board, then solder on the replacement part. Here's a video showing you the process; he's also working on a remote for a car first, so the part you'll be interested starts at about 8:45.

iPad Air 3 Repair - Only charges with cable at certain position + Bonus Benz Fob repair - YouTube

The part itself doesn't look hard to find; one came up on Amazon right away, but you an likely find it on other sites like eBay or AliExpress among others.

Amazon.com: ePartSolution OEM Replacement for iPad Air 3 A2152 A2123 A2153 Charging Port USB Charger Dock Connector Flex Cable USA (White) : Electronics

That should be everything you need to know; you'll need to read through the motherboard replacement guide then watch the YouTube video and decide for yourself if you're comfortable doing the job. If not, you can undoubtedly find a local repair shop that can; this is a repair that most reputable shops should be able to handle with no problem.

Best of luck; let us know how it all turns out!