What about something simple, like "Rehoming Assistant"?
Another thing that might help decide the job title is whether it will be used more internally or also with clients. I feel like if it is mostly going to be talked about within your group, then you have a lot more freedom for a fun, in-joke name or a very generic one. If it is something that clients are going to be told (or given a direct contact with via your website), then I think it has to more clear and specific to the work. For example, if someone calls considering rehoming their pet and your desk staff offer to connect them with a generically-titled "Admin Assistant", it might not be clear to them what kind of help they are going to get, or require the person answering the call to spend more time on explanations.
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Emme Hones
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