Do I also need to do a brainstorming
session for lateral keywords?

Question - We've selected our keywords since we're using the site for an existing consulting business. In a nutshell, we provide consulting services to doctors in a specialized niche market. 

Here's the question. I only did vertical keywords and do I also need to do lateral keywords? 

Answer - I had a look at your keywords and there are quite a few very targeted ones there. 

I'd save all of these to a file like you have done and then do a lateral search as well. You'll probably find a LOT of additional related words that you can then brainstorm as well. 

Save these all to their own individual keyword groups on your computer and you will have lots to choose from as you build your site and in any paid advertising you do in the future.