While some will argue that an image is content, and in a manner of speaking it is, when your web page is indexed and analyzed, the two are completely separate and therefore one cannot influence the other such as you are suggesting. Let me explain.
From the very beginning when Google was a research project, textual content has been indexed separate from everything else. Title tags and link text in particular are indexed separately. Google uses the HTML DOM (document object model) to separate template elements and content as well as non-textual elements. Google indexes within the content index only the textual content. The reason for this is to be able to apply semantics to the content itself. This was somewhat limited in the beginning, however, semantics did exist from day one. Google stores similar indexes for other DOM elements such as the title tag and link text. Google told us so in the original research paper. As well, this very likely includes indexes for image alt text and other elements where Google cannot parse the content itself such as for video. How this is segmented within the index stacks is not totally clear.
It is important to separate the content index from all other elements to purely know what the textual content is about. Otherwise, an excessive use of other non-textual elements can skew the semantic value of the content. In this index, all terms are linked to a term index with positions from the starting point of the content (at least) and very likely other starting points within content blocks (customary in semantic analysis). This allows Google to use these measures to extract phrases and relative importance of terms to each other as well as their specific meaning in how the term is used. In addition, other semantic measures and analysis can be applied such as semantics to determine topic scores which is very important to Google.
Why it is important to separate textual content from other elements is that the content text is not abbreviated and therefore gives better analysis results. For example, any image alt text as well as the title tag and the link text are typically written in an abbreviated style and does not necessarily maintain a sentence structure and therefore does not lend to subject, predicate, and object analysis. These shorter elements generally do not indicate the pages topic as well as the content itself. There is some exception to this. Title tags, because so much attention is applied to them, can indicate the topic better than any other brief textual element. In fact, this was noted in the original research paper and it was discovered that doing parallel searches against the textual content and the title tags as separate searches, weighed, and blended gave a better results. I believe this at least occurs for link text as well. But it is not clear if this is done for image alt text. I believe that it is not done even though image alt text has it's own index stack (something similar to an index only many-to-many join table).
Google will see
Meet the! We are awesome! and CompanyName separately and index them separately. CompanyName will not carry the same weight, value, or potential as the rest of the header tag.
It is possible that when you do a Fetch as Google it may appear as though CompanyName is augmenting the h1 tag, but it is not. You will notice a box around the image alt text. While it is inline, it is separate. Being inline, only indicates the position of the image alt text. Nothing more. It is possible that search queries are made against the image alt text, but it will not carry the same match potential that content will. I do not recall search query matches against image alt text as showing up in the SERP snippets which is a major clue. You will also notice that matches against other elements such as templated elements generally do not show up in the SERP either. You will see the description meta-tag matched as well as content block elements such as paragraphs. I do not recall header tags being matched of late.
You will be subject to the ever changing winds of how Google and other search engines weights various DOM elements by doing this. It is far better to have a steady result than a variable result.
From an SEO perspective, this is a terrible thing to do. I always advise NOT to place an image in your header tags and especially the h1 tag. This is because the h1 tag has particular semantic value over and beyond everything except the title tag and link text. Everything else follows especially traditionally non-textual DOM elements.