The Eucharist for Messianic Noahites
HITVEADUT BIRKAT HAMAZON UHAVDALAH
Why should you do it?
Your Rebbe has decreed for you that this blessing is to be regarded as the toast of the Brit Chadasha for the sake of which he made Kiddush HaShem. Anyone who makes this toast is in the Brit Chadasha. The sages taught us that the blessing over the Kos for Havdalah is guaranteed to merit you a place in Olam HaBa. It is the substitute for Toda'ah offerings in the Beit HaMikdash according to the Brit Chadasha.
It means as long as you give thanks for the Shabbat then you are in the Brit Chadasha and your sins will be forgiven (you are merited a place in the world to come).
Remember when you say ata chonantanu or bein kodesh lekol then Shabbat is over and we are in the Motzei Shabbat Tzom. And when one finishes saying Havdalah over the Kos shel bracha and drinks, the Motzei Sabbath Tzom is over!
So why instead of rushing to eat other food you should eat the shirayim and dregs from Shabbat observance instead?
This is to say to yourselves that you do not chase after the world but we miss the time we spent with the Lord and even though you are now free to pursue other things the first thing we do is try to grab hold of the remnants of Shabbat instead, thereby baring witness that you are not of this world but belong in the Shabbat rest.
In this way the Lord of the Sabbath remains with you as something precious all week.
That is why you should want the shirayim after Shabbat and the blessing over the cup is finished. Because you are immediately putting yourselves in the mode of longing for Shabbat.
To long for the Shabbat is Yesod, and Yesod is the true body of Metatron who is the flesh of God's Name (Ben Adam Kadmon. If you "long for Sabbath all week" (placing shirayim in a Jewish Monstrance) it will help you prepare better for the coming Sabbath and will encourage you to do positive mitzvot!
And if you say Havdalah to sanctify the end of the Sabbath with thanksgiving your offering is accepted and your sins in neglecting the negative mitzvot are forgiven.
This is how to be a Messianic Noahite according to Hassidean Romaniote Minhag.
The instructions for combining Birkat HaMazon and Havdalah over the same Kos are found in the Shulchan Aruch.