Got it working using <message-properties-transformer>
:
<http:outbound-endpoint
address="http://user:pass@example.com"
followRedirects="true" method="GET" exchange-pattern="request-response">
<message-properties-transformer
scope="outbound">
<add-message-property key="CUSTOM-HEADER-NAME"
value="custom-header-value" />
</message-properties-transformer>
</http:outbound-endpoint>