Good question. There are two issues when transferring ownership rights in a copyright: (1) ownership; and (2) recording/notice.
On the first issue, you simply need to execute a contract (often referred to as a transfer of rights) that assigns the rights from you to your company. There can be nuances of this, but assuming you want to transfer your complete bundle of rights to your company and have not previously transferred any rights in these copyrights to anyone else, it is a relatively simple matter. One thing to be mindful of is making sure you execute the contract in a valid form. Have you formally formed you company? Does it have officers? At a minimum, make sure to get the contract signed by witnesses so it isn't just you signing on both parties behalf. This assignment fully effects the transfer of rights.
That being said, there are reasons to follow through on the second issue of recording the assignment to provide notice to the public that your company now owns the rights. Many of my smaller clients have, among others, sale of the company as an eventual exit strategy. Many others are looking investors. Both of these points involve forming a valuation. To get the highest valuation, you want a clear and public record of the company's assets.
In a nutshell, there are many other reasons to record the transfer and almost no downside.
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You fill out and record with the Copyright Office and Assignment. You're correct that copyright applications and registrations (asnd assignment) don't prove ownership, but they do put everyone on notice of what's claimed and by whom.
Note that if the "load of documents" weren't authored by you, or weren't minimally creative, they may not be worthy of copyright protection. You may not be seeking the best form of protection for them, and are best off seeing an IP lawyer to discuss your and your company's situation regarding these documents.
PLEASE READ THIS BEFORE YOU COMMENT, EMAIL ME OR PHONE ME. I'm only licensed in CA. This answer doesn't make me your lawyer, and neither do follow-up comments and/or emails and/or phone calls, and you shouldn't expect me to respond to your further questions if you haven't hired me. We need an actual agreement confirmed in writing before any attorney-client relationship is formed. This answer doesn't constitute legal advice, and shouldn't be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue.