The two main differences between hydrogen and helium are that helium has a much higher mass, and would therefore have much more momentum coming out of the Big Bang, and secondly that helium is a noble gas and so it interacts very poorly with other atoms. Both of these differences would seem to have the effect of making helium far less efficient at colliding and converting its kinetic energy into thermal energy and radiating that heat away in order to undergo gravitational collapse. If helium has two attributes that make it worse at thermalisation and gravitational collapse, you would expect the hydrogen to quickly fall down to form galaxies while the helium stays in orbit outside.
Once the CMB temperature falls to below around 6 or 7K the energy of CMB photons would be too low to excite the lowest energy level of the helium, so I would expect the helium would become transparent to the CMB and be essentially looking out at a universe of zero kelvin and its temperature could fall practically uninhibited at which point you might think it could condense and form superfluid helium droplets which would be invisible to us. I would guess the size and distribution of the Bose-Einstein condensate droplets would depend on the evolutionary history of the galaxy, but you might imagine it could produce the required mass sizes.
So, are there some observations which would rule out helium as a component?