Can I remove a RAR file's (known) password without recompressing the archive?

Out of the box, no, you can not. Version 3 of the RAR file format (implemented first in WinRAR 2.9) encrypts the actual data itself, as well as the file headers (if requested) using AES-128 encryption. With just WinRAR, it is impossible to simply "remove" the password from an archive, since the data itself is encrypted with the password.

You could make a quick batchfile implementing a "remove password" feature, which could simply unrar the archive, and then re-compress the files without a password.

---

Technically, the data is compressed before being encrypted. This indicates that, given enough knowledge of the RAR file format itself, one could create a tool to AES-decrypt the datastream of the compressed files, and then save it into a new RAR archive. It should be noted, however, that this requires extensive knowledge of the file format itself.

Given the number of open-source tools that support password-protected RAR files (e.g. unar), one could learn how to do this by reading existing source code, and then using the decrypted, but still compressed, bitstream to generate a new RAR archive. However, this is far from a trivial task, as you would then have to rebuild the RAR header manually as well (or at least ensure the file format's compatibility).

Actually, I think a previous answer from another question already answers this:

You can do it by using this method . Tools > Convert Archives > Compression > General > Set Password > then click ok and save :)

but it re-compresses the archive.