You can use HistoricFormProperty (history service, createHistoricDetailQuery()), which do contain property ID/value and task id. Make sure the history service is set to full. You can query based on the right process/task and see what value was submitted back then.
I remember to set the history service to full (instead of audit). I suspect that this "HistoricFormProperty" can be found in the "taskVariables" of a historic-task-instance in the rest-api?
Right, you wanted to use the REST-api. Take a look at 'GET history/historic-detail' in the user guide, which allows you to filter out the form-properties for a task/process-instance start form…