A lot of native plants will manage fine in that sort of situation. They can also be quite cheap so it won't be so bad if you have to dig them up later.
Maybe just go for a hedge along both sides of the garden with fences, this will give you some more privacy and still leave space for the dog to run around.
Two plants that are really great for hedges are blackthorn and hawthorn. They look quite similar with masses of small white flowers, but one flowers in March/April and the other flowers in May/June so you get a longer period with flowers.
In the autumn, hawthorns have bright red berries and blackthorns have blue berries (they're used to make sloe gin).
A hedge alternating blackthorns and hawthorns looks great and they grow very easily.
You can also mix in some other species as well, dogwood is often a good choice as well, that has white flowers and very dark, almost black, berries.
Other good species to include are elder with white flowers and dark reddish/purple berries. The flowers are used to make eg elderflower cordial and the berries can also be used to make cordial or jam.
Or maybe something like hazel which has lots of catkins in the winter and you can sometimes get hazelnuts as well.
The cheapest way to buy hedging plants is what is called "bare root". As the name suggests, these do not come in a pot of compost but are literally the plant with bare roots.
These are generally sold November to March/April so you would need to wait until October/November when the nurseries start selling them again.
Bare root plants are going to cost you from less than £1 each up to around £2. In contrast, you're probably looking at something like £8-10 for a plant in a pot.
The £2 plants are often a year older than the £1 plants so you're usually paying for a slightly bigger plant.
Most people say that for a hedge you need at least three plants per metre or to get a more dense hedge then go for five plants per metre staggering each one.
So the cost for a hedge would be anywhere from £3 per metre (3 plants @ £1 each) up to £10 per metre (5 plants @ £2 each)