How do I get to the wall cavity above a double floor joist from below?
Disclaimer: We bring in outside contractors for our electrical installs, so I can give you some general tips, but I'll defer to others that may know more about this specific problem.

In general, joists, and any lumber supporting a load on its side, will lose much of its carrying capacity when it is drilled or cut on the top or bottom.  For passing through the joists, you should make the hole in the center (between the top and bottom) of the joist, and make it the smallest hole that allows the wire to pass freely.

Having a double joist running beneath a wall means that the wall and joists are designed to be load bearing.  So because of this, unless someone with more structural knowledge says it's ok to drill a hole, I'd avoid it.

While I can't think of a good way to get the wire into the wall, you could install your outlet on an adjacent wall, or you could go for a floor mounted outlet as a last resort: