I don't think you can ever do a rigorous ruleset for Bughouse Dominion. There's just no consistent way to parse Ironworks, Mine, Soothsayer...
Hmm… what if you just replace the rule "when you gain a card, unless otherwise instructed, it goes in your discard pile" with "when you gain a card, unless otherwise instructed, it goes in your partner's discard pile"? Cases when you have specific other instructions on where to put it, such as Mine or Develop, don't change; you follow those instructions instead of the default rule.
Does this cause rule problems? Admittedly it makes some cards and some relationships between cards weird.