The builtin Script "Copy Project" can be used with REST:

copyProject.sh
curl -u admin:Welcome1 -H "Content-Type: application/json" -H "X-Atlassian-Token: nocheck" -d @copyProject.json http://jira.server.dk/rest/scriptrunner/latest/canned/com.onresolve.scriptrunner.canned.jira.admin.CopyProject


The Json Payload:

copyProject.json
{"FIELD_SOURCE_PROJECT":"TEST","FIELD_TARGET_PROJECT":"TEST2","FIELD_TARGET_PROJECT_NAME":"A copy of TEST","canned-script":"com.onresolve.scriptrunner.canned.jira.admin.CopyProject"}


See the Confluence Copy Space here