Should I install fiberglass batts underneath my existing blown fiberglass?
Whatever gave you the idea that blown fiberglass is substantially less effective than batts? Though the R-value of batts is 25-30% better, you simply account for that with increased depth. 

The inadvertent gaps and voids that batts will leave you with after fitting around countless roof framing members surely negate any benefit versus simply adding more blown fiberglass. The only thing you should consider doing is increasing the depth, should it be inadequate. 

Also, you can't simply move and replace the fiberglass. To be effective (and not clumped or matted) it needs to be blown in. That means hauling it down to a blower and cycling it through. Not worth it.

I'm in Minnesota (winters routinely to -30F) and blown fiberglass is by far the most common professionally-installed attic insulation. Cellulose is probably more common for homeowner jobs, but they're similar in the end.