It’s possible to pass custom variable parameter via query params to trigger build with overridden values: https://${BAMBOO_INSTANCE}.internal.atlassian.com/rest/api/latest/queue/${PLAN_KEY}?bamboo.variable.some_var=some_var_value.

This method is not described on this page: https://developer.atlassian.com/server/bamboo/bamboo-rest-resources/ . However it is described here: https://docs.atlassian.com/atlassian-bamboo/REST/6.10.2/