Consider the way those accelerator rounds are constructed and you'll be amazed that they group at all. It's basically a 55gr .22 cal bullet in a plastic cup. Even if Remington can get them to leave the muzzle consistently (which I doubt) there's still the mid-flight separation of the cup & bullet. No way does that cup come off of that bullet the same way every time. If you can get a 4" group at 100yds with accelerators, you can probably shoot.
To the best of my knowledge, you can't get accelerator-type bullets for reloading. Some bullet makers do offer 100 and 110gr bullets, though the 100gr bullets are generally considered 'plinkers'. Hornady's 110gr V-Max might be a good place to start.
If you really want the ballistics of the accelerators from your Handi-Rifle, I'd recommend a 22-250 barrel.