Two ideas:

Have an option while sniper hunting to open your other eye instead of just the one looking through the scope.

When one shoots closing an eye actually hurts your accuracy. That's IRL logic, but it would be helpful in game too for knowing where you are pointing your gun so you don't lose track of your target that got a little too close to see in the scope anymore and also to know how far away said target is.

This could be something like this:

Another Idea I had is to remove the sniper hunting reload minigame based on timing and make it to where you have to press a key combo instead to make it more similar in aspect to a bolt action rifle or lever action. like press shift +e -> shift +Q -> shift + f and then you can fire again.