You can really do everything under one domain. So I would register the domain and use WordPress. Then for the "blog" portion of your site you can create a category called "Blog". Add that category to your Menu. Now every time you write something for your "blog" you make it a post and assign it to that category. Now every time someone clicks "Blog" they will get your latest blog posts.
No need to have two separate domains. That's the beauty of WordPress you can have static and dynamic content on one site.
I think you may be getting tripped up on the words "website" and "blog" when they are essentially the same thing. A blog is a kind of website where content is posted by date and category. But you can also create static pages and add them to your menu and they can link to other content, including your blog section.
Here's an example. Look at my blog http://blog.2createawebsite.com. The "What's New" tab is my "blog content". So you could label yours "Blog". Notice when you click, you get my latest blog posts. But when you click "My Tools" you get a static page. This is an example of blog and static content on ONE website.
My next podcast is going to be on this very subject because I think a lot of people get confused.