I'm not a teacher, but although it SHOULD be "figger-it-out", you will most likely hear "figgeridout" a lot, especially if you come up here to the Southern Hemisphere and hear English in NZ or Australia.
I think this term is originated in America so we have to pronounce it with an r.
But why do we care for where the r is, because I think when we pronounce it, the sound is the same?