That’s great that you can get a link from Gov website but the link will be in the footer and this means every page of the website will be linking back to you as normally websites share the same footer across the website.
Google like natural links and when you will use a key phrase to link back to your client’s website (especially in the footer) this will hurt you as Google can easily guess that the link has been created to manipulate SERP results.
My opinion, if you are linking from footer is to get link on brand name instead of a keyword this way you will get the link juice. I am not saying you will be able to achieve your goal but still you will increase the website authority which will help you in different ways.
You also said that you don’t have lot of unique content for each page. There is no shortcut or pass here you have to create unique content for pages in order to increase your chances to rank for different county related keyphrases.
Honestly, this strategy in my opinion is not good and Google can easily guess what you are doing and reward you accordingly.
I think the better idea is to get a link with a brand name to increase authority, create unique content for county pages. Get some quality links from different industry related websites and you will rank better in SERPs from the targeted key phrases.
Just a thought!