Btukfyl

If one makes a bracha on something not Kosher (meat and milk,chicken and milk, pork…) should he eat it to avoid saying G-d's name in vain or not?

He already said God's name in vain since non-kosher food doesn't warrant a blessing. Eating the food now isn't going to help that.

אכל דבר איסור, אף על פי שאינו אסור אלא מדרבנן, אין מזמנין עליו ואין מברכין עליו לא בתחלה ולא בסוף. (שולחן ערוך או"ח סימן קצו:א)‏

If one ate something prohibited, even if it was only prohibited rabbinically, one does not combine him to a zimmun, nor would he say a beginning or after blessing [on that food]. (Shulchan Aruch OC 196:1)

After such a mishap one should say ברוך שם כבוד מלכותו לעולם ועד.

According to the Kitzur Shulchan Aruch, in the event of saying a Bracha l’vatalah (an unrequired blessing), the fix is to say, “Baruch Shem kavod Malchuto l’olam va’ed.”

There are also some who suggest making the blessing, “lamdeni chukecha” after that.

In either case, one should not consume the non-kosher food.