Much of the logging doesn't come from rest assured itself, but the underlying http client library. I customized the output of that apache client library by including a logback-test.xml (the name is important, that's how logback knows to use it if it's on the classpath) in my src/integration-test/resources/ folder.
AFAIK Apache client uses Log4j2 and you have to add a bridge library to intercept the normal Log4j2 calls and route them to logback instead. I looked through my Gradle configuration files and I didn't see any bridge dependencies, so I'm not sure exactly how the log files from the apache client are being routed into logback unless rest-assured comes with its own built in bridge dependency.
Bridge dependencies
I've attached my logback configuration file so you can see which parts I silenced.
Now that I've read your question again, how are you "capturing" this output? The "correct" way to do what you want would not be to use a custom library, but to use a xml configuration stanza with no console appender, and only a file appender.