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Serberys Raiders get a normal move at the start of the first battle-round (Skirmishing Line). AdMech also have a Doctrina for +3" movement, also selected at the start of the battle round.

Is it possible to first use the +3" doctrina and then move the Serberys Raiders 15" (instead of 12") with Skirmishing Line?

At the start of the battle round, if there is a DOCTRINA ASSEMBLER model from your army on the battlefield, you can select one Doctrina Imperative that has not yet been active for your army. If you do so, until the end of the battle round, that Doctrina Imperative is active for your army.

Skirmishing Line: At the start of the first battle round, models in this unit can make a Normal Move. They cannot end this move within 9" of any enemy models.

I haven't read up that much on the new book but from what you posted I don't see why not.

As the active player, the AdMech player could choose to activate their Doctrina Imperatives before or after the Skirmishing Line depending on which is favourable.