he pretty much goes in a different spot each time.
Why not train him to go to one specific place each time? If he's going all over the yard, why bother having a nice yard?
But to answer your question, how much fertilizer are you putting out? Granular or liquid (hose end, etc)? How often? Not sure that I am clear on the 4 Step process. You don't fertilize four times per year? I can't see that a pound or two per thousand feet is enough to get on him. I agree that once it is irrigated in, there is no need for concern.
As a professional landscaper, I commonly put out several bags at a time (over a couple of days, as time permits) in ornamental beds. I usually do this by hand if the beds are too small for using a spreader. Frankly, I'm old school and I can sling it out by the handful about as accurately as most folks with a spreader. I've measured the area of the beds and weighed the fertilizer. The point is, I sometimes wear gloves, usually not. My exposure is much greater than your puppy's, and considering that I have somewhat sensitive skin, I have never had a problem.
Oh, you asked about your puppy. Well, we used to use bean beetle dust with arsenic for flea powder. Dust'em good, let them run and roll in the grass, and lick it off their paws. Never had one sick. Everybody knew that you didn't mess with my dad's kids or his dogs. Both were important to him and he would never do anything that would endanger them.