Thanks for the feedback, everyone. They've been really insightful. I should've mentioned how I went to eforms and typed up the letter based on which state you're from (NJ in my case). I think @Anna Sagatelovareally hit the nail on the head:
Here's what I currently have on my letter:
"Thank you for submitting your rental application to us. We regret to inform you that your application as been rejected due to one or more of the following reasons:
[REASONS LISTED ON THIS THREAD]
Your application may have been denied on the basis of a report given by a consumer reporting Agency and/or on the bases of the information obtained from other sources.
If your application has been denied on the bases of information obtained from other sources, then you have a right under the Fair Credit Reporting Act to make a written request within 60 days from the receipt of this notice regarding the nature of this information. As per deferral law, we are prohibited from disclosing the source of this information.
If your application has been denied on the bases of information obtained from a consumer reporting Agency/Agencies, then you have the right under the Fair Credit Reporting Act to make a written request within 60 days from the receipt of this notice to obtain a free report from any of the following Agencies:
[...]
You also have the right to dispute your credit report on the basis of its inaccuracy or incompleteness. You may have other rights under Consumer protection Law or credit reporting for which you may contact your state Attorney General’s office or your local Consumer Protection Agency.
Important Note: The Fair Housing Act is a federal statute that prohibits discrimination in the sale or rental of housing, as well as in residential real estate-related transactions, such as advertising, mortgage lending, homeowner’s insurance, and zoning. The law make it unlawful to discriminate on the bases of race, color, religion, sex, and national origin, disability, and familial status."
So the above letter may be used for disqualifications related to credit. Other than that, I should just stick to writing a simple "Thanks for your application, but we've found someone else" (or some other reason), as many of you have suggested. Thanks for the help, everyone.