IMO that's an extremely skimpy budget. I suggest that someone wanting to thru-hike the trail set aside something like $4000, and that's for, say, 150 days or less.

What does the $300 include? In particular is transport to and from the trail already covered? How well could you deal with any sort of unexpected event --- injury, sickness, equipment failure?

Just buying food and nothing else, $300 for 30 days (and if you walk fast you'll surely do more than 300 miles in 30 days!) implies $10 per day for food. Fortunately, prices in the south are lower (definitely give up on this idea farther north on the AT). But you're going to want a shower on occasion. You'll feel somewhat deprived if you never ever eat in any sort of restaurant.

A related question is whether you already have all the gear you need (including guidebook). Any gear adjustments can quickly eat up funds.

I can't respond to the sleeping bag question as I started the AT on the early side, when a 30F bag definitely wouldn't have been sufficient. Search and/or post on whiteblaze.net to get responses from folks who have experience with that part of the country.

My feeling is that you would do well to start perhaps in April, and get good at mining hiker boxes. You've got a big advantage in starting from the very beginning of the trail at the time when many other hikers are just starting --- because people start the trail with a lot of body fat and a lot of fear, so they leave food behind in hiker boxes. Unless you're pretty picky you could keep your food cost down substantially that way. Just talk to folks that know (start out by asking the folks at Mountain Crossings) where the best hiker boxes are along the way.

Being in warm weather helps too, less need to get off trail and get warm/dry when ambient conditions are pleasant.

So I'm not trying to talk you out of this, and given the idea of "go until you run out of money" --- lots of people do this (some planned, some unplanned). Just do leave yourself enough $$ in reserve to do real basic stuff like get a bus ticket at the end and be able to eat en route home!