Or a small breed like a Mini Aussiedoodle!

Easy to fix..adopt a dachshund!

But if you plan to leave it in the yard all day while you're at work, adopt two of them, so they can have company and play together. Many dachshund mixes have short legs also and would do as well.

15 Awesome Dachshund Mixes You Won’t Believe Are Real [holidogtimes.com]

More doxie mixes: [k9ofmine.com]

Try youtube , selfhelp for advice on holding the drill ect .
You'll need a 2 foot or 4 foot level to keep the battens straight up and looking professional .
You might want 2 drills , or not , depends on hardness of wood .one with proper size drill bit for pre-drilling some holes and one with screw- driving bit .
Maybe some string to guide cross pieces .
Tape measure .
Pencils .
Possibly a chalk line
And something suitable to cut stuff with .
Screws of a proper length .

Or keep the dog in the garage

My sister had near enough the same problem. There were no rules from the rescue place, but the dogs could get over the front wall. What she's done is put a flower bed between the lawn and the wall and fenced it off with chicken wire type stuff. So the dogs can't reach the wall without scaling the wire fence which is too flexible for them to jump up and over. I doubt it would satisfy a strict rule though.

Sounds like you might have an easer time having planks the right length delivered, then rebuilding your fence. Had to rebuild ours after a storm took out most of it a couple years - it wasn't that bad. Your plan would functionally work, but I don't think you'd be pleased with the aesthetics of your updated 2-tone fence. Oh, and a compression drill will make the work go so much faster! Good luck!

Now it's time to consider dating a landscape Gardner or a carpenter..???

I have no idea what a batton is and neither, it seems does Google.

Perhaps it would be easier to remove three feet of earth, thus lowering the level of your garden by the required amount?