Answer
Report Designer ships with its own bundled JRE, located inside its own install folder (for example, under a 'java' subfolder of wherever Report Designer itself was installed) — use that bundled JRE specifically rather than trying to install a separate standalone JDK/JRE, and make sure JAVA_HOME is configured to point at it. Pointing the environment at the correct bundled JRE from Report Designer's own installation directory resolves the 'javaw not found' error.