Yup, if the fixed head isn't flowing enough then there's the problem. If the concealed shower is overly restricted then it's not getting the flow to activate the pump switches as I think you've guessed by now. If the fixed head has small water holes too then there may be insufficient head to overcome the restriction, does it work ok if you remove the 'fixed head' from the outlet pipe?
Universal (negative head) pump could be the solution.
Universal (negative head) pump could be the solution.