Possible but I think the speed needs to be a touch faster. & you have good endurance but the 32 could be a touch faster with the 1600.
I was 4:25/9:24/16:15 1500/3k/5k at a time. Pretty similar looking to where you're at. Gonna be tough the make huge strides towards sub-16 without being 4:30s for the mile & sub-10 2 mile.
I wouldn't get too hyper-focused on an arbitrary barrier & just listen to your coach. You'll eventually break 16:00 & will improve with more miles & time in the sport. Building to 55-60 miles is good place to start. Keep to what your coach prescribes over the summer. You don't want to burn yourself out for the actual season if you start training like a sub-16 runner now. Mileage, strides, and tempos, will pay dividends once your season starts.
FWIW, you still have two months or so where you can train like an 800-3200 runner. See if you can chip away at your PRs in there and then take a couple of down weeks before getting up to 55-60 miles in August. Find some folks to do a couple of time trials with.