The gemara says that we compromise in saying ברוך שם כבוד מלכותו between what Yaakov Avinu did of saying it loud and what Moshe Rabbeinu did (not saying it at all in the parsha of shema in parshas vaeschanan). Therefore, we say it, but we say it silently. Our custom on Yom Kippur is to say Baruch Sheim... out loud because of a special status we have. The Maharsha and Tzlach both point out that the issue with saying Baruch Sheim out loud is only a problem when being used as a break between the pasuk of Shema and V'Ahavta. Therefore, if one is not saying Shema, or is only saying the first pasuk of Shema Yisroel, they can say Baruch Sheim out loud.
This approach fits well with Rashi who explains that R. Meir in the Braisa considers כורכין את שמע to mean, without pausing between the pasuk of Shema and V'ahavta. Therefore, R. Yehuda argues and says that even if you are mafsik with a pause, it is not sufficient, because you need to be mafsik with the phrase of Baruch Sheim. Based on this approach, there is no question that the function of Baruch Sheim and the discussion of saying it loud or quietly is due to the hefsek between Shema and V'ahavta. However, Tosafos explains the problem with being כורכין את שמע and not being מפסיקין, doesn't refer to a hefsek between shema and v'ahavta, rather a hefsek between שמע ישראל and the word Hashem. According to this approach, there is no discussion at all regarding hefsek between שמע and ואהבת, therefore when R. Yehuda comes to argue on R. Meir and insists on the saying of ברוך שם כבוד מלכותו לעולם ועד, it would seem that after saying shema there is an inherent reason to say the pasuk of baruch sheim, not merely for the purpose of serving as a hefsek. Therefore, when the gemara concludes that we compromise to say it quietly, it may apply even in a situation where one is just saying the pasuk of shema and not the entire parsha, they should still say baruch sheim quietly and not out loud.