Thank you! upreferred simplifies the units perfectly.
Now my examples become:
julia> 1u"M" * 1u"L" |> upreferred
1 mol
julia> 1u"L" * 1u"1/cm^3" |> upreferred
1000
A related post recently came up with the same issue: Make Unitful simplify expressions like 1μm/1m to a unitless value by default