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This does appear to be a mounding moss.

Yes. I think you already have an idea of what the problem is here. You will just have to stop your dog from urinating on the lawn. It is being burned by ammonia from urine.

That will be the only way that it will recover.

Cool, wet weather favors moss growth. If this describes your spring, do nothing and let warmer weather take care of the moss. Moss can also signal poor air flow and crowded plants that don't let sunshine penetrate. Those you can fix by adjusting your gardening practices. There are chemicals which reduce moss but are not for use on veggie gardens. The following article has more detail: https://www.gardeningknowhow.com/ornamental/foliage/moss/killing-moss.htm

It looks like Swan's-neck thyme moss (Mnium hornum).

Don't take the bark off when you are removing the moss, it can open the tree's growing layer to a pathogen infection and wood decay.

Mosses need little if any fertilizer. In fact, fertilizer can burn moss while encouraging weeds. The pH of soil can also help or hinder mosses. Many species thrive in highly acidic soils (pH below 5.0), so by lowering the pH with powdered sulfur you can encourage mosses while discouraging weeds and lawn grasses that generally prefer a sweeter soil. For the most part, mosses will do best in a damp, shady location.

https://www.gardeningknowhow.com/ornamental/foliage/moss/growing-moss-gardens.htm