Below are examples of three permissive licenses worthy of your particular attention. More broadly, however, bear in mind that in general, permissive licenses come with obligations. Almost all require attribution—that is, including the copyright with the software. The BSD 2-Clause License Simplified, for example, makes it very clear that usage requires attribution. It’s not heavy lifting, but organizations need to manage it.
Removing copyright information from anything is certainly problematic (at least under U.S. law). Title 17 of the United States code Section 1202 explicitly forbids removal of copyright information. Practically, the downside risk may not be that high, but Chapter 12 goes on to talk about civil remedies, statutory damages, and criminal offenses and penalties, all of which make this nonlawyer uncomfortable.