Why is that a problem? Boosts in different directions don't commute, so in deriving the full group of (linear) inertial motion transformations for 1+3D spacetime, one must take account of ordinary rotations...The bad news is that the group is non-commutative, so making two boosts in different directions becomes a problem.
The group law assumption in all these types of derivations boils down to the hypothesis that boosts along a fixed direction should form a 1-parameter Lie group.
Maybe some of it was known even earlier. The standard references are Ignatowsky and also Frank+Rothe (though I've been unable to source them, and they appear to be in German anyway -- I haven't found a translation).Did you see the 'note added ...'. Apparently this was all known in the 1920's.