Q. During the omer; on Shabbos (during the summer), when taking an early Shabbos one can be faced with 3 mitzvos simultaneously- counting sefira, bircas hamazon and reciting the shema
what is the proper order of operation?
Shkoyach.

A. Poskim disagree in regards to precedence when krias shema and birkas hamazon are an equal option, since both of them are Biblical mitzvos. Sha’agas Aryeh (21 ”“ 22) maintains that reading the shema is a more frequent obligatory mitzva and should preempt birkas hamazon (todir veino todir). Although one could eat more than two times a day, it is not mandatory to do so, however, the reading of the shema is.
Similarly Rivavos Efraim (O.H. 8: 477) rules that one who began eating early on Erev Shabbos should read shema before benching. He adds that medakdekim or people who are careful and particular in their mitzvos, habituate to stop in the middle of the seuda, when the time to read shema begins.
However, Mishne Halochos (15: 101) quoting Rav Shulchan Aruch (299: 3), that once we are sitting eating seuda shelishis and night began, we don’t stop and read shema then, as is obvious from the Shulchan Aruch itself. He argues that the same should apply to Erev Shabbos also, when receiving Shabbos early.
Horav Shlomo Miller’s Shlit’a opinion is that one does not have to stop the meal to read the shema, rather recite birkas hamazon first.
In regards to the counting of the omer, most Poskim maintain that the Biblical mitzvos of birkas hamazon or kerias shema take precedence over in our times Rabbinical mitzva of sefiras haomer.

Rabbi A. Bartfeld as revised by Horav Shlomo Miller Shlit’a