Based on her personality, she'd probably highly recommended bringing her along. Which sounds arrogant, but it's mostly because she knows first hand how dangerous these monsters can be and doesn't want to see anyone be hurt because of them. However she also understands that she can't do it all herself, so if there are others who want to complete the quest she would respectfully let them take it without objection.

She is capable of more than hunting beasts, it's just what she's good at. So there won't be any problems if others want to take those quests. She'll probably insist on coming, but whether she does or not depends on the Guildmaster's decision. I imagine there might be times when Chizuru thinks it'd be best for the group to take on the task themselves without Shalquoir babysitting them.