You're right, however....
Looking at penalty 3 in rule 9-1, it's clear that the ball is awarded via the arrow. However, POI makes more logical sense to me in this situation.
If the simultaneous violation occurs when more free throws are to be shot, those free throws are shot and play continues as normal. This actually contradicts 9-1 penalty 3, since we are supposed to penalize everything in the order it happened. Since the violation happened after the foul, penalty 3 should be enforced. 9.1.3M(b) also goes against this principal, since the BI occurred after the foul. The case play says play continues from the free throw as normal, however, inferring this double violation actually goes to POI.
I think the penalty in 3 should indicate POI rather than AP. That would reflect the way it's actually called on simultaneous FT violations (next shot or AP if no more shots are to follow).
This would affect the hypothetical in the OP, as well as the situation where, on the 2nd of two technical foul or intentional foul free throws, there is a simultaneous violation. As the rule reads now, by the letter, you would go to AP in this case instead of granting a throwin to the team who was fouled.