Would it be possible to start with the option to have a large construct companion at 1st level.
It could sacrifice a little for the option, but would enable builds involving riding the construct right away.
My quick suggestion, make large construct have the following changes.
+2 STR, +2 DEX replace their base attributes. (basically a -1 to each)
The bludgeoning unarmed strike stays the same, but the slashing/piercing option drops a die step.
Note that although the construct has a lower STR, it should technically be able to carry more, as the bulk of medium and smaller items is reduced.
The choice of base can be retrained later for free when Advanced Construct feat is acquired. (so the penalties can be retrained out of, when their construct's frame more easily supports the large size natively)