I'd get it in ASAP. I'm not sure how quickly they are calling people, but figure you'll get a phone call within a few weeks, to set up video interview. Knock out the video interview mid-late August. Have in person interview early September. Assuming you can get the time off to come up to PSM? Depending on how much they've already filled for October, you might be able to work out that class. November class for sure I would think.

When you get that first phone call he'll lay out a potential timeline for you and what the earliest class is they are trying to fill that you could be in. If he says September class, then you could inform him that you wouldn't be available until October 1st due to prior commitments with your current employer. I don't see the process happening quick enough at this point for you to be in before the October class anyways though.

At any rate, get it in so you can get the process started. They won't see your honoring your commitments as a negative if that's what you're worried about.