I am not familiar with every run-out method ever devised for a weak notrump doubled. However, most that I have found require playing at the three-level if Responder desires to show both minors. A two-under forcing pass may do better.
1NT-X-P = suggests run-out to a minor. Opener responds by bidding:
2C = club preference
XX = diamond preference
If Responder started with a one-suited minor hand, he passes 2C with clubs, bids 2C after a redouble with clubs, or converts the redouble or 2C call to 2D with diamonds. If Responder started with both minors, he passes 2C or converts the redouble to 2D.
Responder could also pass with a minor-major combo. With, for example, 4H/5C, Responder could pass and then either (a) pass 2C or (b) convert redouble to 2H to show four hearts and longer clubs. I'm not sure that I would do this, but it seems like a plausible move.
Responder might even opt to convert the redouble for penalties. If, for example, Responder felt that 1NT XX'ed was a good bet if Opener prefers diamonds but would prefer to play 2D otherwise, then heck -- pass and pass.
If Responder has values, instead, he redoubles himself. 1NT-X-XX. Otherwise, systems on.
Maybe this works; maybe not so good. But, it's a two-under forcing pass option, and I like that as a theory discussion.