How can I rebrand my 5-year-old company?
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It sounds to me that, as a niche, specialized, service industry company that this may be less of a threat to you than it first seems.
Your "brand" is so much more than your company name--it's your interaction with clients, your service levels, responsiveness, quality and specializations.
Your brand is the collection of all the impressions that you leave with your customers every time you interact with them. That brand matters--the company name is just one element, that in business-to-business markets can be much less important.
With a giant competitor, your size may well be an asset. Can you respond faster? Meet customers' specific needs better? Just be easier to do business with? That's what will help you continue to bring more business.