My 2 cents. Be very careful about charging "pet rent". I think I read somewhere that "pet rent" paid can be a defense by tenants when it comes to deposit deductions for pet damage.
I'm in CA and here is what I do to protect me and my clients when I/they think about accepting pets:
In your advertising, say "pets may be considered and are subject to owner approval"
That gives you the control to decide how many and what kind and under what terms you would accept a pet or pets.
Advertise the rent in an amount that you would accept for folks with pets. See my concern above about pet rent.
Increase the security deposit as needed and legal to make you comfortable with the pets. If they don't want to pay it, they can go elsewhere. We usually increase the deposit by $500-1000 per pet. If you are not comfortable with the pets, decline to approve them.
In California, deposits cannot be non-refundable, so plan on refunding all the deposit each and every time. That will help you make good decisions regarding accepting applicants.
In CA I like to call ALL deposits "security deposit". Tagging deposits for other uses, pets, cleaning, rent, etc., in CA limits the damages for which a LL can withhold. For example, a LL cannot apply a last month rent deposit to pet damage or outstanding utilities and visa versa. Calling the deposit security in CA gives the owner the most leeway to apply deposits to monies owed from whatever damage source.
If approving pets, require the tenant carry renters insurance and name you and your PM as additional insured. Again, if the tenant doesn't like it, they don't have to rent from you.
Require that all pets be licensed and up to date on their shots. Require proof.
Check with your insurance company regarding whether any dog breeds will raise your insurance premiums or cause your coverage to be declined. Don't rent to folks with those breeds.
Have a very strict pet addendum.
Check with WA regarding treatment of deposits, pet rent and requirements for insurance by the tenant. There may be pitfalls there.
And finally, check the jurisdiction in which the rental is located to see what, if any, restrictions there are on the number of pets in a household. Some cities in Silicon Valley cap that number at 2.
I hope this helps. I'm not an attorney and can't give you legal advice, but with over 25 years of PM experience in a "tenant-friendly" state, I've learned how to protect myself as a landlord and property manager in addition to my landlord clients.
If you are worried about these particular folks and their pets, say no.
On the bright side, I often find that responsible pet owners (check references) make stellar long-term tenants. That could bring you piece of mind for years to come.
Good luck!