I'm familiar with all of the technology involved, but I'm not sure about the applications you're describing.

With a Have-A-Heart, the specific goal is live capture and release. There is no killing involved. The animal might be properly freaked out at the experience of being trapped, but that is specifically so as to permit an animal's live relocation.

With a bolt gun, it's meant to be used in a slaughterhouse scenario, which is a whole moral discussion of its own, but at bare minimum one wants the animals to be kept as calm as possible until the bolt gun is applied, because stressed out meat tastes worse than calm and placid up until the moment of death.

With hunting, the goal is to kill the target as cleanly as possible, preferably with a single bullet. That's the Scenario A I'm describing above.

If one were hunting an animal with the intent of killing it, then a trap, followed by a knife or bolt gun, would maximize the terror felt by the animal to be killed. Sure, one may be putting less lead out in the environment, but at the cost of putting the animal through... almost the most appalling experience of death possible, with the admitted exception of a poorly-aimed bullet or arrow, followed by a wounded flight through the woods and slowly bleeding out.

So... if one's absolute maximum goal is to reduce environmental lead, yes, that is one way to do it, but the moral implications of that method seem pretty rough.