You can still do both. There is just a 10-15 minute window where Harriet won't let you do business with her after you've killed some animals. Just get your business done with her first and then go out hunting.
You have to keep pumping the animal with sedative rounds until you you see the red X as if you've mortally wounded it. Instead of dying, it will run for a little bit and then pass out. You can then walk up to it and take a sample and if you wish, revive it for extra naturalist xp, provided you have some animal reviver sold by Harriet. The bigger the animal, the more rounds it takes to sedate it. You cannot sedate birds and small critters like squirrels, rats, chipmunks, toads, frogs and bats. The shots will kill them.