Usually the meaning of דמשק אליעזר is translated as Dammesek Eliezer, meaning that he was from the city Damascus in Syria as in Bereshis 15:2 Targum Onkelos and Radak. וַיֹּ֣אמֶר אַבְרָ֗ם אֲדֹנָ֤י יֱהוִה֙ מַה־תִּתֶּן־לִ֔י וְאָנֹכִ֖י הוֹלֵ֣ךְ עֲרִירִ֑י וּבֶן־מֶ֣שֶׁק בֵּיתִ֔י ה֖וּא דַּמֶּ֥שֶׂק אֱל...

Since shechita is not required would that prohibit bringing a ben pakuah as a korban? Or would you just schect it anyway and continue on?