A synopsis of the genesis of the C.
Bell Helicopter proposed upgrading UH-1Bs with the "540" rotor system. This modification was made to about four-to-six UH-1Bs in the U.S. The results were promising enough that additional "540" rotor heads were shipped to a very few units in Vietnam. They didn't receive the full UH-1C retrofit: "540" rotor head, new engine, new powertrain, gears, and rotorshaft, bigger canted tail, bigger tail rotor, asymmetric horizontal stabilizers; just the "540" rotor system.
The first UH-1B/UH-1C hybrid was a plain UH-1B with the "540" rotor head only. It was discovered that in Vietnam's "hot and high" flying environment, their performance was marginal, and they could exceed the tail rotor torque limits. Also, they were too heavy for the UH-1B's T53-L-5 960 shp engine.
The second type of hybrid was the UH-1B with the "540" rotor, T53-L-11D 1100 shp engine upgrade, powertrain, and tail rotor. These modifications improved performance, but with the increased torque from the more powerful engine, and larger tail rotor, the tail had to be modified by adding the camber and increasing the width. These modifications were incorporated into what became the UH-1C model. One unit which flew UH-1B/UH-1C hybrids was the 174th Helicopter Assault Co.
As to why the museum has its facts wrong I don't know but am in contact with them to right matters. N9-3104, which the records you quote to be a C, is in fact a B, as recorded on both its data plate and its civil registration VH-NVV papers. The Navy never possessed any C's.
BTW, I flew the Navy Hueys for a total of 675 hours and put in a few hours on the C in Vietnam.