What is the optimal method of dealing with leaking brick veneer?
I have brick veneer too, but I have a 1” air space between the veneer and stud wall with plywood covered with a sheet of plastic moisture barrier. This allows any moisture that enters the wall to drain down the 1” void and out the weep holes at the bottom of the wall. 

It would be nearly impossible to remove the interior drywall and install a correct system like mine. 

Rather, I’d invest in verifying that you do not have any cracks in the mortar joints or between the mortar and brick and then install a clear sealer on the brick veneer. 

I’d first hire a masonry contractor to “point” all the mortar joints. Then get the best clear brick sealer with the highest amount of silicone available. (I think mine was 5%.) and hire a painter to install it. They will spray it on and you should make sure they “flood” the brick to the point of refusal. They’ll use a lot, but it’s worth it. 

The clear sealer will last 5-6 years, depending on the weather. Be prepared to re-do regularly.