Can't get 2N3904 transistor to allow the collector current I want, when used as a switch
The gain (\$I_c/I_b\$) of a BJT is lower in saturation than in forward active operation. A typical value used to define the onset of strong saturation is \$I_c/I_b=10\$.

So choose the resistor for your "?" location to give 13 mA or more into the base of the transistor.

Also notice that the \$V_{ce(sat)}\$ and \$V_{be}\$ for this transistor increase above our usual crude estimates for them when operated at currents above 50 mA:



You might have more success by choosing a beefier transistor.