In Shulchan Oruch Orach Chayim Siman 487 Sif 4 it is brought the minhag of saying Hallel in shul the first night of Pesach
בליל ראשון של פסח גומרים את ההלל בצבור בנעימה בברכה תחלה וסוף ובן בליל שני של שני ימים טובים של גליות
On the first night of Pesach, we complete the Hallel with the community with a pleasant tune and a blessing at the beginning and end. The same is on the second night outside Israel.
In an answer giving here it's mentioned a number of reasons for this custom. One was that it was to be motzei those that can't say Hallel themselves. Another reason was because of the fact that we don't make a bracha on Hallel during the seder and making it in shul takes care of the bracha. However we find a few reasons why we don't make a bracha on the hallel of the seder. One is because of the hefsek that we make in the recitation of hallel, another is because hallel is said on the night of pesach "b'Toras shira" and not "b'Toras Kriah". Therefore this reason seemingly wouldn't satisfy the reason on why we make a bracha on Hallel said in shul.
Therefore I ask, why does the machaber himself say to make a bracha? What is the reason for the bracha (even we find Gaonim that already gave a good reason why not.) Where is he getting this from? Which (earlier) poskim?
(I've checked already in the Tur and Beis Yosef which don't speak at all about this halacha that is mentioned in Shulchan Aruch. I also looked at the Bar HaGolah that doesn't add anything of importance in regards to the source.)