Can you post a photo? Or describe in some detail how much space you have for planting? And is this for sun or shade?

This is one corner, so i want to plant something around each of those corner

What dimensions are you looking for? There are tons of evergreen shrubs and small trees.

Would it be an idea to first put some wire mesh in the gaps to prevent small dogs etc (very small dogs) from leaving or entering? Later, when some nice plantings are present, it will be more difficult to do this.

As Dave says, there is so much choice (I presume. I know nothing about South Carolina). What sort of thing do you like?

When a choice is made, be sure to give them enough room -- don't plant too close to the fence. Better to err on the generous side in my opinion.

I thought of adding wire mesh but I think the neighbor will still freak out😭. Plant wise I really don’t know much and will probably want something small that doesn’t grew much roots and end up it the neighbor’s yard. I thought of getting some garden fence to put along the property line, then plant something along that area either in planters or in the ground directly. Thanks for all your input.

P.S. We had to redo the fence, moved the post and removed the board that touch his fence, that’s why I think put wire there would make him angry too. Alternative is to build a whole new fence along the property line that’s unnecessary if we can find a way of keeping small animals from coming into our yard.

It might be an idea to walk around in your area to see the sorts of things that do well there, and the things you like. South Carolina looks to have a very different climate (humid subtropical) to the one I live in (cool temperate).

I feel you will like something rather fast-growing.

Climbers in pots in my opinion are not such a good idea as most climbers (the ones that I know) are vigourous and won't be happy in a pot. And I suspect your neighbour might not be happy with a climber from your place peeking around to his place. Might be better to choose an evergreen of some kind, something that doesn't loose its leaves in autumn.

I've just had a little look on google and wonder if Baccharis halimifolia would be a good choice.

https://www.gardenia.net/plant/baccharis-halimifolia

a fastigate barberry in that corner should do the trick. So is Neighbor intending to squeeze into the five inch clearance to maintain his fence?

Thanks @tete_a_tete & @kitasei2 for the feedback.
I have no idea how the neighbor intends to maintain his 5in, just waiting for Spring to see if he will abandon it and we have to take of it if we don’t want weeds/tall grass in our yard.