First of all, discount the time for a rebuild suggestion. You could be running oil through a worn out engine like poop through a tin horn and that isn't going to make it miss sometimes and not others.
Since it does it under accelleration, I'd be looking at ignition rather than fuel. Whacking open the throttle increases combustion pressure which can snuff out a weak spark. First of all, install new plugs if you haven't already. Gap them to Harley's specs but not over. The wider the gap, the more energy needed to initiate a spark.
If you have a multi meter, check the resistance in the resistor cap (plug cap). If you get more than 20,000 ohms, it's time for a new one. I like to see 12,000 or less myself. Also see if the plug wires are touching metal because of the possibility of leakage through the old wires. Does it run worse or start hard when the weather's foggy or wet but run better when dry? If so, that's a sign that the old plug wires need replaceing.
However, I expect it to be something else. My first guess would be the ignition coil. It has some age on it and has lived in a pretty hot environment. Heat eventually caused the insulation in the wires inside to break down, letting it short out. Sometimes you can find little cracks in the outside plastic insulation, but other times they look fine. I can't remember if Harleys have one or two coils, but I bet that's the problem.
In the late '70s and early 80's Kawasaki had similiar coil problems with their early pointless ignitions, especially the coils roasting under the tank on the hot running 1300's. They got so hot that Kawasaki had to put insulation on the bottom of the fuel tank because you could open the lid and see the gas boiling at the bottom of the tank.
If you can, get good high power coils like Dyna. If you can get something with more get up and go than the OEM, you can increase the spark plug gap .005 or so and that will make it run better, especially on startup. If you get aftermarket coils, make sure the primary resistance is correct for your bike. Most Dynas are 2.5 ohm which is a common requirement for electronic bike ignitions. There are coils out there with more like .8 ohm resistance, and if your bike needs 2.5 ohms, you'll need to add a resistor of around 1.7 ohms between the positive lead and the positive terminal on the coil.
It could be a corroded connection somewhere between the coils and cdi box, but considering the age and mileage of the bike, I'd suspect the coils first. Even if the OEM coils are ok, new high powered coils will make the bike start and run much better.
There's good mechanics out there, but too many of the young guys don't spend enough time reasoning out the problem and are too quick to start changing parts. The backfiring under hard throttle should have been a tip off that it's ignition and not carburation. If you can, find some old fart like me that's been around the block several times or some older long haired dude that grew up wrenching on the pre-evolution jobs. Most likely he's had all sorts of breakdowns and failures. If you need more help, email me. Good luck, buddy.