It's possible to define an a different faction that's player-type and can be selected there, but if all you want is the same rules but a different set of units I suggest just having an alternate ark with alternate build menus.
You'd need to suppress the code that grants the free starship constructor, however, because otherwise it could still build the normal starships for you.
And actually I think you'd run into difficulties with all the planetary controllers also having the default build menus (space dock, starship constructor, turrets, etc), so having an exclusive unit set would be tricky (involves defining an alternate Controller and making sure it gets used).
So you might actually wind up needing to define an alternate Scenario and wire up the start-game button to use that instead of the normal one.
But I'm guessing that you can accomplish your immediate goals just with an alternate ark.