No, you may’t forged a card with insanity via mayhem.
Usually, discarding a card causes it to enter the graveyard instantly:
701.9a To discard a card, transfer it from its proprietor’s hand to that participant’s graveyard.
A mayhem forged is enabled if that card was discarded this flip. It doesn’t matter that the insanity mechanic redirected the discard into exile, it nonetheless counts as discarded:
702.35a Insanity is a key phrase that represents two talents. The primary is a static means that features whereas the cardboard with insanity is in a participant’s hand. The second is a triggered means that features when the primary means is utilized. “Insanity [cost]” means “If a participant would discard this card, that participant discards it, however exiles it as a substitute of placing it into their graveyard” and “When this card is exiled this fashion, its proprietor might forged it by paying [cost] slightly than paying its mana price. If that participant doesn’t, they put this card into their graveyard.”
Nevertheless, solely the cardboard in exile counts as discarded, and insanity forces the selection of casting it or placing it into your graveyard. Whether or not you forged it or transfer it to your graveyard instantly, the cardboard now within the graveyard is a brand new object that was not discarded, and doesn’t have mayhem enabled.
400.7. An object that strikes from one zone to a different turns into a brand new object with no reminiscence of, or relation to, its earlier existence.
