Questions like this always raise red flags about "balanced" construction, although this isn't a typical veneering situation. My concern would not be about grain direction of your two plywood layers, but about whether both sides of this door are the same, i.e., will there be a 3/8" layer glued onto the inside core as well? If not, your construction needs to be robust enough to counteract any warping or stressing you are setting up by the unbalanced construction. I can't think of any way to calculate this likelihood, unless you want to try making a sample. Keep in mind that, with a garage door that has sun exposure or differential temperature conditions (large difference in indoor vs outdoor), you need to consider these factors in your testing as they could affect the moisture exchange or deforming tendency of this construction.
Sorry, can't be definitive. My hunch is that you will be able to get away with your plan without major problems. I just wanted to suggest that you consider things that could be factors. If you can do this without actually bonding the two plywood layers together, I'd feel more confident about it.