Is it possible to mail the Unlock Key with the mailing function of the License Manager?

Yes, it is possible. The XML provided by License Manager for transformation contains "Unlock Key" in the @UnlockKey attribute of the Version/LicenseType/Purchase element. So you can use following XSLT construction to include it into the message:
<for-each select="//Purchase">
Unlock Key: <value-of select="@UnlockKey"/>
</for-each>