I'm very late to this party, but given the nature of the letter in which BLARINGNAMECORP's name was to appear—namely, a letter in which you seek acceptance into a university program—I think you made the right call in casting the company's name as Blaringnamecorp.
In this regard, it's important to note that the all-cap treatment of company names in logos, letterhead, advertisements, and Web pages is in the vast majority of cases a marketing decision, driven by people in the company's advertising/marketing/publicity departments. In many cases, if you check the copyright information on the home page or "about us" page of a company that presents itself in the style BLARINGNAMECORP, you'll find the official name of the company given less stridently, as Blaring Name Corp. or BlaringName Corp. or even Blaringnamecorp.
The point is that the all-caps presentation exists to promote brand recognition and sales. It's aggressive to the point of obnoxiousness—at least that's how some readers will take it, if they view it as the equivalent of someone shouting at them. I certainly wouldn't want to take the chance of offending the university admissions officer who was reviewing my application.
U.S. copyright law requires that people writing about copyright-protected products and entities indicate those entities' special status by treating the terms as proper names—an obligation that traditionally was satisfied by using initial caps for proprietary names. The law does not require writers to replicate all-cap treatments, cutesy superscript characters, gratuitous punctuation marks (as in "c|net"), brand-associated color schemes, brand-associated special fonts, or anything of the sort. Why? It isn't because the company wouldn't like free reinforcement of its marketing efforts; it's because lawmakers recognized that special visual treatments are merely promotional, not intrinsic to establishing ownership and copyright.
For 19 years I worked at computer magazines that reviewed thousands of products for many hundreds of companies. Our policy was to initial-cap company and product names, but not to reproduce all-caps, embedded punctuation, superscripts, font styles, colors, or any other special treatments that the companies had tricked out their product names with; the policy further dictated that we not include ©, ®, or ™ symbols with product names.
Our rationale for the policy was that we were in the business of appraising the products, not training readers to salivate when they saw the marketing-approved form of each name. In any case, we never received so much as a letter of complaint from any company's legal department for enforcing our policy: As long as the product name was initial-capped, the proprietary nature of the company name or product name was deemed sufficiently established under copyright law.
This strikes me as a common-sense interpretation of a company's rights and a writer's obligations with regard to proprietary names. An admissions officer is put on sufficient notice regarding the copyright interests of Blaringnamecorp if you style the name that way; I can't see how anyone's legal interests would be justly promoted by a rule requiring you to mar your application with multiple blasts of BLARINGNAMECORP.