Where exactly are you located? And how much room do you have to plant (bed depth) and how tall do you need these plants to be/grow in order to have the privacy you desire?

I will say that conifers tend to be one of the ideal choices for this sort of situation so you are ruling out a lot possibilities right there. But they are not the only solution.......just need to know where you are to offer other suggestions.

Fire bush is the common name of a shrub native to very southerly locations and is not commonly grown in the PNW. Burning bush (Euonymus alatus) grows here, is not considered invasive here, but is deciduous and seldom achieves enough height to be effective for privacy screening.

Bark-resistant plants will be hard to come by as dogs will bark at things they can hear, even if unseen.

Jumping, pawing excited dogs will spoil the planting if not excluded, you will have to do more than just plant something between them and the neighbor.

I started a boxwood and lonicera nitida hedge last year in (amended) heavy clay soil, and both types of plants are doing quite well. The Lonicera Nitida is the faster grower of the two. Both are broadleaf evergreen, but they're not going to get that tall that fast, and they're unlikely to ever get much taller than 8' from what I understand. How tall of a hedge do you need? I have a camellia in the back which also looks to make a reasonable evergreen hedging plant, but again it's quite a slow grower.

I'm not hugely fond of coniferous hedges myself, so I do understand, but if you want a broadleaf evergreen hedge you'll need to be willing to wait quite a few years for it to grow in. And you will need to keep the dogs away from your new hedge until it's well established. Luckily my two dogs aren't fence runners, but they're quite adept at trampling plants in other parts of my garden.