From a pumping level of 50’ a 7 GPM, ½ HP should produce 7 GPM at 70 PSI. Even if the water level pulled down to 170’, that pump should still deliver 7 GPM at 17 PSI. Recovery rate doesn’t mean anything until the water level is pulled all the way down to 170’. If the water level doesn’t pull down more than 50’, the pump is the problem, not the well.
A 10 GPM, ¾ HP pump will deliver 7 GPM from a water level of 170’ and still produce 40 PSI. If the well makes enough water to keep the storage tank full when sprinkling, then you just need a well pump that makes enough pressure for the gun, and the extra pump and storage tank would not be needed. If you let me know how much pressure the gun needs, I can tell you what size pump to put in the well so the storage tank and booster pump are no longer needed. No reason to pump the same water twice if it is not needed.