Would it help SEO to put only keywords inside header tags and style other text beside it to look like part of the header?

Moving my answer from Stack Overflow where your question was off-topic

This is a very unusual suggestion - I've never once seen any of the SEO experts even hint at something like this. It sounds like something that used to work years ago - a bit like keyword stuffing - and could be considered to be a "black hat" these days.

It certainly doesn't meet Google's (admittedly vague) guidelines - they say you should write content and mark it up for the user, not the search engines. It also doesn't meet any of the standard conventions for writing web pages.

As for negative impact:

• If Google decides this is black hat, they will penalise the ranking.
• If they don't consider it black hat, they still might consider the page over-optimised for that term (depending on the rest of the optimisation), and penalise it
• Even if they don't penalise it, it means the page is optimised for that phrase only and not for for local or long tail searches.
• Google is putting on a lot more emphasis on LSI these days, so repetition of the same keywords is not particularly effective.

As I said, I've never seen this technique even suggested, never mind recommended. Based on all that, I personally wouldn't use it when optimising a site, but then I am clearly more cautious than your client :)

I don't see any reason that breaking the titles like this could boost SEO in any measurable way.

The general rule of thumb for SEO is that if it makes no sense to have your page set up some way and it goes against web standards, it's probably bad not just for SEO but for your site at large (you've already discovered that it's bad for accessibility). Good SEO is all about choosing from multiple valid, standards-compliant ways to set up a page.

In one word the answer would be NO.

But adding keyword in header helps user to understand about the topic/page. For helping in SEO you must follow Google's guidelines.

As @FluffyKiten said:

It certainly doesn't meet Google's (admittedly vague) guidelines