It's hard to reconcile him "begging" to be picked up with him then being in a biting "frenzy." I tend to think you're misinterpreting his "begging." I think Eclectuses are not hands-on birds, and that they can be feisty. Search You Tube for the Riley the talking Eclectus videos. He doesn't strike me as a bird who wants to be held and touched a lot, and he gets pretty feisty at one one point, throwing around his owner's keyboard.
I think the advice often given here to just put up with biting and not "reward it" is just wrong, unless you're a conducting a psychology experiment or want to get your bird so he can be a star on TV. Similarly, I don't think there's any value to the practice of labeling your bird as "dominant" and then trying to assert your own "dominance" over it.
Why must things devolve into a struggle for power of "dominance"? My theory is that many people are very afraid of the amazing power of their birds, which is very real in the larger birds, and thus focus energy on constantly trying to "dominate" so they don't feel afraid.
Instead, you should be using ways to distract your bird to stop the biting, such as throwing him off balance by twisting you arm when he bites or is bending down to bite you. ("He goes to the floor," as Birdman666 puts it, although the bird doesn't need to actually fall off your arm ... the point is to interrupt the behavior.) Just letting him bite you repeatedly is going to instill a pattern of behavior in him. Avoid the situations when he tends to bite ... instead, enjoy his presence, sit near him, talk to him, get a stand, spray him with mist ... all good stuff that won't get you bitten. See what develops over time as you get to know each other rather than getting into a huge power struggle with him, which is probably he cause of the biting.