My current novel has three main characters, Josie, Liv, and Lue. What I worry about is if Liv and Lue's names are too similar. Liv is short for Olivia and Lue is a dog. I really like both the names, but not if it's going to confuse my readers.
If you are worried that they might be, then they probably are. My test is to try and put both names in a sentence, or a few, and read it out loud. Having said that, in real life, people with the same or similar name can be friends, so it happens.
I think the reader can keep a dog and a person straight regardless of similar names. It's when the names are similar between two similar people that it gets harder to follow.
Readers don't know the differences between the two as instinctively as you, and might get the wires crossed. It's best to give one of them a nickname that won't cause confusion, like 'Olive.'
I don't see a problem here. You shouldn't have to bend over backwards to design your characters' names to make them painfully different. I once knew a guy named Bob who had a cat named Bub. That wasn't a problem. I recently read a story that had two characters named Steve. The author just called them Big Steve and Little Steve whenever there was the possibility of confusion. It's perfectly okay to write your characters the way people are in real life. Readers are smarter than we think. They'll deal with it.
I don't see a problem. The names are different enough. Plus, it's a dog and a human, people will be able to tell them apart in your story; unless Lue is a talking dog or Liv starts barking, I think your readers will be fine.
I'd say write the names as you've got them now, if you like them. After you're finished, get a few people to read your story. If your readers get confused, then it's probably a good idea to change one of the names, but with 'find/replace' functions on nearly everybody's writing software these days, that shouldn't be hard to do. You could change 'Liv' to 'Livy.' Both are nicknames of Olivia, but Livy looks different on the page.