And even if its density increases while you cool it (does it?)
Yes, it does, that's what every gas does, you can easily check it just by playing with the ideal gas equation.
However, helium condenses at around 4K, so - even assuming ideal gas behavior - we won't get more than about 12 g/L. That's actually pretty close to the density of a liquid helium, which is most likely a good estimate of the highest density of the gas. But it is still less than the density of the gaseous UF6 (13 g/L).
Your number for liquid helium density is in wrong magnitude and the assumption of ideal behaviour is not a good one. Helium at atmospheric pressure is the least ideal gas because it has the lowest critical pressure, and helium 3 is far less ideal than 4. If Earth atmosphere were very slightly denser, helium 3 would be a permanent gas.
Helium 4 at 2,2 K and its then vapour pressure of 0,05 bar has density of 146 g/l. It is not much increased by further cooling.
Compressing helium 4 to 30 bar while cooling it to 1,8 K compresses helium 4 to 180 g/l while still a liquid.
Warming helium 4 under its vapour pressure to 4,2 K, where vapour pressure is 1 bar, allows it to expand to 125 g/l.
On further warming, the critical point of helium 4 is at 5,20 K, where pressure is 2,27 bar and density 69,5 g/l.
Expanding the gas, at 1 bar and 4,2 K the vapour density is 16,9 g/l. So more than 1/8 the density of liquid - no wonder considering the proximity of critical point.
So helium gas near its condensation point is very nonideal gas, and shrinks to a much bigger density than an ideal gas would.