There is a maklokes haposkim in regards to if amira lakum is muter bmakom hatzrich/mitzvah/etc if what you are telling the goy to do something which for the yid would be psik reisha. For those that are makil what is the tam/savara/lomdus that there is no “psik reisha” in regards to the goy and that for him we look at the action as being muter, therefore muter for you to tell him to do it bmakom that we are matir amirah l’akum (I hope I have explained this clear enough.)
Many authorities explain that there is no prohibition in telling a non-Jew to perform an action that is defined as a pesik reisha. See, for instance: Rema 253:5 and 337:2; Magen Avraham 253:18 and 41; 276:10 (clearly relating to a de’oraisa prohibition); 277:7; 314:5; 337:2; Tehillah Le-David 336:4; and many others.
Although some prohibit (see Peri Megadim 253:41), the Mishnah Berurah (253:51) thus rules that one can be lenient for needs of Shabbos.
The reason for the leniency is that the problem of telling a non-Jew to perform a melachah is a problem of shlichus (making him an envoy), and for a pesik reisha the shlichus is being made for a permitted act (Shulchan Aruch Harav 253:10; Yeshu’os Yaakov 253:14; Tehilla Le-David above).
Another explanation is that the shvus of telling a non-Jew is more lenient than a regular rabbinic prohibition, and therefore it is permitted in a place of a pesik reisha (Magen Avraham 314).
A discrepency between the two rationales emerges when the Jew has specific intention for the melachah of the pesik reisha to occur, though he asks the non-Jew to perform the non-prohibited labor — for instance, if he wants the groove in the ground, but asks the non-Jew to drag the bench.
According to the Magen Avraham, this might be prohibited (because the Jew’s intention is the groove), whereas it will be permitted according to the Shulchan Aruch Harav (however, see Avnei Nezer (210:8), who writes that this is prohibited even according to the Shulchan Aruch Harav).