It takes experience to know what sacrifices you, individually, can make for six months. That's a lot of what we learn on the first long hike.

With only $2k you'll be making all the sacrifices and then some, and more when you get back home. Most people wouldn't call this enjoyable, will quickly adapt: have their fun with the money they have, then go home early. There's nothing wrong with that shorter experience.

But, there's probably something wrong if your plan is to be totally miserable for six months. Some people want that, even need it for a time. But that's a minority of AT hikers, and six months is just masochistic.

Fixed Costs: Can't cheat much here

$1k + transportation, enough to get a place and job when you get home.

$700-800 for gear, and go to /r/ultralight to figure out how to get durable, cheap, light, and safe at the expense of research time, sometimes minor gear modification.

Variable Costs: duration dependency

$2k @ $1/mi for food. Ramen, PB, fried chips.

$200 @ $0.10/mi for incidentals: shower, McDonald's, Uber, hostel

It's going to cost a fixed $1.7k just to get away and hike without totally screwing yourself.

Every $110 you save beyond that is another week on the trail.