Is the plug and harness at the back of the alternator in good shape? Might want to look at it closely, a good connection here is important. Unfortunately, the rebuilt units from the chain stores don't have a good track record on a 6.0 powered trucks. While they might work ok on a gasoline powered sedan, the load on them is pretty high with the diesel.
After going thru a couple of alternators myself, I took one to a local shop for repairs. After returning it the first time they fixed it by installing some different "heavy duty" parts and it's been working correctly for a couple of years now. Like yours the indication was an intermittent BAT light on the dash.
"Thinkin ain't knowin"
2006, King Ranch, F250, Lariat, Super Duty, Powerstroke, FX4, CC, SRW, 20in Ultra Motorsports Goliath wheels. 2in leveling kit, Husky 16K roller hitch, ScanGaugeII. Sinister coolant filter. Blue spring. BPD EGR cooler. Riffraff CAC boots w/metal cold side tube, STC update. Fumoto valves. Rotella ELC. DashBoss. '08 Mirrors. Mishimoto Rad and 200° stat. High Idle Mod. 3G Alt.