Understanding Noise Figures for BJT transistor
It's never a good approach to add a resistor in series with the source to get to a point where the device noise figure is better.  All you would be doing is increasing the noise form the source so that the device noise is relatively less.  The total noise at the output would be increased.

Can't you operate at ~5mA where the device noise is better and it would work well with 50 ohm?

I don't know about how the device noise changes with Vce but you could use a biasing circuit that doesn't lose any of your available supply voltage.  For example don't use an emitter resistor but instead use DC feedback from the collector to base.  With such low impedances you should be able to do that without adding significant noise in the bias network.