Where I am, Canada, but it's probably similar in the states, all wires have to 1-1/4" back from the face of a framing member to avoid being hit by nails/screws. Mechanical protection can be used if they have to be in harms way. Hopefully your subpanel feed is armored, or I suspect it won't pass. Think about it, miss the strapping by a little with a drywall screw and your gonna be well connected to the panel!
Drilling holes in the joists is no big deal, just keep them near the middle so they don't weaken the member.