It would be better that if specifically Christian, Muslim, Jewish, etc., prayers are banned from military or civil functions, any other prayers, including ones to some universalistic deity, should not be recited. Additionally, Christians would be well-advised to use the power of the purse and boycott said hospital until its management relented on its rule prohibiting specifically Christian prayer by chaplains.
It appears you have created an intractable problem that can only result in no prayer at all being allowed in a multi-religious environment. Most chaplains speaking to a multi-religious audience know they are personally praying to their specific god, they know where their heart is, but the speech is such that listeners can apply it equally to their faith. Some apparently want to use the opportunity to proselytize, which is rightfully not appreciated by the institutes that hired them to cater to the needs of a multi-religious group.