I was just learning the Mishna Berura and Aruch Hashulchan on this topic, and one thing that seemed to be strangely taken for granted by the Mishna Berura, without any source for it that I can find, was the individual's (even in a minyan with a Shliach Tzibbur) recitation of the word "Emes" at the end of Shema. The Aruch Hashulchan seems to consider an opinion which agrees that the individual shouldn't say it because it would lead to 249 words instead of 248 since the Chazzan repeats the words HE"E. However, when the M"B mentions this issue, he seems to only consider the rectification of leaving out the first Emes for the Shliach Tzibbur alone, while the rest of the Tzibbur would seemingly be left with 249. I must be misunderstanding the M"B. Either way, like the title of this question, I'd also like to know, what is the source for the halacha of adding on Emes anyway? The midrash that is brought in the Mishna Berura seems to only address the repetition of the three words HE"E.?
To address a point of terminology that may clear things up somewhat: the word "emet" is not really being "added" to shema at all. Rather, it is the first word of the blessing recited after shema i.e. emet veyatziv (or, at night, emet ve-emuna.)
So, why are we particular not to pause between the end of shema and the first word of the blessing emet veyatziv?
The source for this is the mishnah in Berachot 2:2:
רַבִּי יְהוּדָה אוֹמֵר, בֵּין וַיֹּאמֶר לֶאֱמֶת וְיַצִּיב לֹא יַפְסִיק.
Rabbi Yehuda says: Between VaYomer and emet veyatziv, one may not interrupt.
(See the gemara starting at the bottom of 14a for an explanation as to why.)
So the individual has to recite emet immediately following shema. (By the way, I don't think Aruch HaShulchan entertains a possibility of the individual not saying emet; like the Mishnah Berurah he seems to me to only be referring to the shaliach tzibbur.)
Now, how does such an individual get to the desired number of 248 words? I agree that this gets confusing, but it seems to me that both Aruch HaShulchan and Mishnah Berurah understand that the "emet" recited by the individual does not count towards the total, as it is part of the next blessing. On the other hand, the emet recited by the shaliach tzibbur (or one of them if he recites it twice) does count, presumably because he does so for the purposes of the getting to the correct number of words.
אמת is the first word of the bracha after Shema. The halacha to say that bracha is given in the Shulchan Aruch in 66:10.