Hello Alan!
Welcome to Quality Assurance 101.

You are already thinking correctly in that no one wants a product with dog hairs on it. But, I feel you should have a different mindset on how to resolve it.
Two quick stories:
1) I once received a heat applied sample from a well known supplier who many use products from here. I did not like that I saw long dog hairs on the backing of one of the materials.
2) I once saw a video online last year of a
DTG contractor in Raleigh, NC that I was going to contact for pricing. While the video was running, a cute little cat walked across the work area and I left the site quickly.
IMO, both of these things are very unacceptable. Some people find it cute to show pics of animals in the work area, I don't. I dare not (knowingly) do business with someone who keeps product around animals and would be upset to get a shirt that a cat has possibly sat on before shipping. I have allergies and cat dander is not my friend.

Animals walking across a business video, regardless of where the business is operated, I find unprofessional. I can understand dust/dust particles when product sits for lengthy periods but animal hair is a no-no.
With that being said, you need to put a basic QA plan in place and take it seriously, even though you work from home.
Simply put - do not have your product around your pet and vice versa under any circumstance. It may help to seal your materials in something in case your pet accidentally gets in your product room. Eyeball all of your product before usage given what has currently happened.