@Holly Jackson I don't charge pet rent. Move to Portland instead!
But in all seriousness, the points you raised about your dog in your initial post were compelling. I wonder if you had quotes from other people who know the dog along with their phone numbers as references (essentially saying the same thing about your doggie as you did), if that would be even more convincing than you simply listing her positive qualities?
When I was a renter (in California and NYC and Oregon), I put together a "pet resume" with a photo, vaccination records, vet contact info, and references from previous landlords. I've also brought my dogs TO the landlord so the landlord didn't have to displace herself but could still meet the dogs (I've successfully rented houses with 2 cats and a lab/heeler mix and a pit bull, so 4 pets).
Renter's insurance is a requirement for my tenants, whether they have pets or not, but especially if they have pets. I don't want to be liable if someone's dog bites someone on my property.
I do ask for a $200/pet extra refundable deposit. I don't ask for "pet rent" because I just advertise that pets are allowed and price my units accordingly.
You'll probably have much better luck when dealing with the landlords directly, rather than property managers. Also, at least in Portland, most of the big new apartment complexes going up allow pets under 25 pounds. They tend to be more expensive than a 30 year old unit from a mom and pop landlord, but you also get all the amenities that go with the big apartment complex.
When you talk with a landlord in person, I suggest already having your pet resume on hand, and maybe even having your dog in the car, in case the landlord would consider your pet after seeing the resume and meeting you, but wants to meet the pet. This strategy works best if there is a group showing rather than the landlord setting up a specific time to meet with you and then finding out that you have a dog.
For the few rentals that advertise pets are OK, set up notices on craigslist and other sites to text you right away as soon as those ads come online. Then *CALL* them right away, talk about your qualifications as a renter (including mentioning that you're a former homeowner and take care of rentals as if they're your own home, and that you have several months' rent already saved, but don't bother mentioning your dog if the ad says pets OK), and at the showing, if you like the place, offer to put down both all the deposits AND first/last at that time to show your commitment. If you would pay more per month for that particular rental, e.g. they're asking $1200 but you'd pay $1300 and think it's worth $1300, you could offer to pay that amount instead. I have heard of renters in tight markets offering to pay more for rent than is advertised.
I wouldn't personally offer to pay a year's worth of rent all upfront, because that would raise red flags with me as the landlord.
Hope this helps. If you move to Portland, get in touch! :)