No, that's a bad idea. You want each page to be set for it's own keywords, not a site-wide keyword.
It will take some time but to do the SEO the right way you want to optimise each page on your site for different keywords. There's always a lot of variations that you can get into any area, and in most cases the page title will be what you want to base the keywords around. This will give you the most changes of getting good rankings because you're not competing with yourself for the same keyword on every page. You also need some variety or else there's a good chance that you'll hit some filter that Google has either now or in the future that looks for duplicate H(x) tags over the site. They look for a huge amount of factors, so I'm pretty sure that they'd be onto that already.
You say that you don't want to add something other them a H2, but that's not a good arguement. All you need to do is style the H1 tags the right way and they'll look the way that you want them to. Use the H1's for the main page titles because that's what they are meant for.