The role of an H1 influencing rankings has certainly diminished, but I would still say it's worth having a keyword in there.
What it sounds like to me is that you're designing your page around the user first, and SEO/H1s second and that's exactly how it should be.
Now, in theory, the higher up you can have the H1 tag the better - as search engine crawlers do crawl linearly from top to bottom - and it is has been said that the crawlers sometimes do not crawl the entire page. For that reason, you'd want it nearer the top as possible.
But I'd certainly not want to skew the flow of the site just to force an H1 tag to appear higher - certainly not in this day and age where the H1 is less of a ranking factor. I'd be confident that by having it in there somewhere in the of your site is more than good enough. If you want, visit the site that's linked in my profile - we actually use an H1 tag that's lower down the page as well and not had any worries over it thus far.
Hope this is some use to you!