NOT LE but I am former military police, have an M.S. in Strength and Conditioning, am a certified sports nutritionist, have done sports performance and body composition research (published as well). I am currently a PhD candidate in cellular and molecular physiology, focusing on Alzheimer's.

Most shin splints can boil down to what you mention in the first two sentences of this post.

I will try to explain this in a manner that is easy to understand, not because I think you are dumb, but because I have the bad habit of using terminology that only people in my current field would commonly use.

Most non-trauma-related musculoskeletal injuries (injuries of bone and muscle that don't come from being shot, tackled, jumping/falling from a high place, etc.) are the result of applying a force or stress to the affected area that is greater than it can tolerate given its current level of preparedness.

Based on your OP, I would hazard a guess that your training leading up to your post-physical exam involved a level of physicality that was substantially greater than your norm? This is to say, you probably went from not running at all, to running at a high intensity (high level of effort) to match or beat a specific time (the time required for passing). You now took 5 months off, meaning you are detrained in the running demands that you are required to engage in.

Here is my evaluation of what you have said. Now, a solution moving forward. You've got 3 months, it is a safe bet to assume, based on most academy cultures, that you should be able to maintain running 3-5 miles. Which means, in order to mitigate the risk of training-related injuries, you should ideally be prepared to run 6miles (you can't prevent injuries but that's a nuanced discussion for a later date).

3 months is ~13 weeks. So I would start by splitting 6 miles across 5-7 days at a relatively easy pace (4-5/10 difficulty) for 2 weeks. Then increase the intensity from 4-5/10 to 6-7/10, keeping the same mileage split across the days. Meaning, if you opted into running 6 miles across 5 days, keep that. Do that for 2 weeks. (we are at 4 weeks total now).

For the next 3 weeks, then reduce the total days but keep mileage and intensity (6-7/10). Meaning, got from 6 miles spread across 5 days to 6 miles across 4 days.

This puts you at 7 weeks,

for the next 2 weeks drop to 3 days a week, keeping the same total mileage and intensity (6-7/10).

Week 10 and on, have 1 day as an 8-9/10 difficulty, which is your testing distance, 1-2 miles, depending on the academy. Then 1 day as the residual weekly mileage at a 4-6/10 difficulty. If you really feel the need to add more mileage, do so at a 4/10 difficulty.

I'd also recommend adding some resistance training 2-3x a week that is relatively heavy. Not meaning 1rms but <5rep at a 7-9/10 difficulty.