Thinking about this a bit more, I realize I gave answers without much explanation.
The buoyant force arises from the displacement of the surrounding fluid - in this case air. So the important point is the volume of air displaced. If you displace a smaller volume, you get less buoyant force: displace a bigger volume, get more buoyant force.
The only role for the He is to keep the air out. As I said, ideally you'd have a container of nothing, displacing 64 m3 of air, giving you 80 kg of buoyancy. But that container would have to be very strong to keep the air out and so very heavy. Then you'd need more volume to support the container. If you fill the container with something at the same pressure as air, then the container does not have to be strong - just airtight, like a balloon. But whatever you fill the container with weighs something and again you have to increase the volume to get extra buoyancy to support that. So you fill it with the lightest thing you can, Hydrogen gas. He gas is second best*.
If you compress the He, you just make it more dense and so heavier for each m3, then you need more volume to support the extra weight of He. You also might need a stronger balloon, which would be heavier!
The only advantage I can think of for compressing the He, might be to preserve buoyancy as the He leaks out. If the balloon were relatively inelastic, as the He leaked* the pressure inside would fall and the density would fall. If the resultant shrinking of the balloon was small enough, the net buoyancy could stay the same or even rise, until the He pressure fell to atmospheric, when the balloon would start to be squashed.
Incidentally, if there were no atmosphere, it would not matter how much Helium or Hydrogen you pumped into a balloon. It would just get heavier and however big the volume, generate no buoyant force. Hydrogen and Helium do not provide lift or buoyant force: they just push the air out of the way and that is what provides the buoyancy.
Edit: * One reason for using He is that it leaks a bit slower than H.