There has to be some type of government involvement in order for bitcoin to have a value in relation to a nations currency.
This is false. Do you think shoes only have value because some government decreed that they have value? What about gold? Did it have no value until some early ruler decreed it should be used as a unit of barter? Of course not.
Bitcoin's value is a function of its scarcity (the fixed supply and impossibility of counterfeiting it), and demand for it. That demand is based on the trust people have in it, and the utility for bitcoin for various functions, as a very efficient medium of exchange and store of value. Government has nothing to do with it. Governments can attempt to
distort the value of something through legal decrees that the value of something is X, not Y (the value established by the free market), but they cannot come up with a more "real" value than Y. Once the government's influence ebbs, the price will revert back to Y accordingly. The same is true for an organized crime cartel, economic monopoly or any other group attempting to distort the market value/price of something.