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.
There is nothing wrong with that. What has developed in contemporary America is a sort of state religion that is not Christianity, Judaism, Islam, or any other faith, but a generic belief that syncretizes some common ground, such as the existence of a deity, the acknowledgment of right and wrong, and some sort of afterlife. In effect, it is not unlike the state religion of the Roman Empire. The empire tolerated the worship systems of all the nations under its rule as long as they all acknowledged the divine nature of Caesar. Two religions did not acknowledge that rule: Judaism and Christianity. The Jewish nation was forcibly dispersed and Jerusalem destroyed in 70 AD. Christianity was persecuted with varying degrees of intensity in the first three centuries of its existence.
The only reason this is an issue is because the government, which as George Washington put it, is neither reason nor eloquence, but force, insists upon public events, such as military gatherings and legislative assemblages, where the state religion is acknowledged in nonsectarian prayers. This state religion replaced the generic Protestantism that prevailed at government assemblages in the 18th and 19th Centuries, and hung on until about 1950 in a diminished form. That generic Protestantism in turn replaced the specific denominations, such as Anglicans and Puritans, that dominated the assemblages of the colonial era.
The best solution for lessening or eliminating the state religion of a vague theism and ethical code would be to eliminate those circumstances where such prayers occur. It would hardly hurt my feelings if Congress, the state legislatures, and city, county, and school district councils dispensed with ineffectual prayer just before they pass new schemes to separate us from our money, property, and rights. At least the legislators would not be hypocrites. Abolishing public schools would also be a good idea. Let the private sector handle this matter, and let people choose schools that adhere to their religious beliefs, or lack thereof.