First, I don’t recommend embedding the slogan in the image. The text won’t be indexed by search engines, and it won’t be read by screen readers used by sight-impaired visitors.
As far as your image size, since you are using a darkened overlay over the image, you can have him compress the image quite a bit. The overlay will hide any softness introduced. It should also be sized to about 1200px wide.
Part of the struggle you’re having is you’re using the Mountains theme, which is designed to have a full screen height image at the top of the website. That’s ok. You can add some CSS to get it sized the way you want.
It’ll be easier to help with the CSS once we see your next image. You can read through this previous thread about the same issue:
In that thread, Doug (@teefers) provided CSS that would also work in your case. You may have to tweak it a bit for your image and slogan.