AFAIK there isn't a special headless version of Java 7. But you can easily tell ordinary Java to run in headless mode. For example:
java -Djava.awt.headless=true com.acme.project.Main
For more information on headless Java, read this Oracle page:
I would need this version to keep it as slim as possible.
Your only options would appear to be:
Ask Oracle if they can provide you with a special build (for money!!!).
Attempt to create your own stripped down build from the OpenJDK sources.
Beware that the Oracle binary distribution licence forbids you from distributing a JRE produced by "cutting down" a normal one. I don't know if this applies to a JRE that you have built from source code. Read the relevant license agreements carefully.
But my advice would be to just use a standard JRE installer.