If your coder/designer is good at what he's doing then it really shouldn't take him that long at all to customize the blog and add it directly to your website.
Hosting WP does require that you have an available database on your server, which could cost extra depending on what your hosting company offers you in the package.

Personally, I think it always makes more sense to host the blog directly on the site - it's better for SEO, makes more sense in terms of keeping your visitors directly on your site and honestly isn't that difficult to install (Can be done is less than 20 minutes). Customizing it to match your website shouldn't take very long either.

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