Raising the soil ph on the lawn
With your lawn the soil test showed your soil pH on the lawn is pH 5.9 which is slightly acid. This has likely been caused over a number if years by the application of high nitrate fertilizer which have lowered the soil pH to a level at which the moss thrives and the grass plants are struggling to access the nutrients in the soil.
Although the finer grasses which make up a finer lawn are traditionally grown in a slightly acidic soil your soil pH on the lawn is too low and application of a handful of lime per square metre will raise the soil pH enough to help the grass compete with the moss and reduce the amount of moss.
We link below is a lime safe to use with animal and dogs etc once applied and watered in if it does not rain in a couple of days it will be safe for the dog to use the lawn