Down here we can cut privet down to the ground and they still shoot from the stump and the roots.
You should be able to chop to bare wood and still expect good response.
Formal upright looks nice and looks easy but in actual fact is extremely hard to achieve proper formal upright. I would look for informal upright because that is much easier to achieve.
The style becoming known as 'informal broom' or 'natural' style might also be OK with this tree.
Is this large leaf privet or small leaf species? Nothing for scale is evident in these photos so it is hard to tell what size the trunk or leaves are but good shohin will be very difficult with the large leaf privet because of leaf size and longer internodes. Small leaf privet is OK for shohin size bonsai.
