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When it comes to the history of the nusach of the first Bracha of Birkas Hamazon, where do we find the first source of this pasuk being within it? Are there earlier sources that don’t mention it?
I am looking for early sources that present the nusach of the first Bracha, both with and without this pasuk.
In terms of early Rabbinic sources: Rav Amram Gaon's, Rav Saadia Gaon's, and Rabbeinu Shlomo b'Rabbi Natan's texts omit Ps 145:16; Machzor Vitry includes it in only one manuscript (the London manuscript, which has many [more than usual] additions and alterations, and which was the basis for one of the major printings); Rambam does include it (see here, 55).
Of the popular Rishonic commentaries on the siddur, Rokeach and Abudarham do not mention it, while Yehuda ben Yakar does include it. Kol Bo writes (siman 25) that Moshe wrote the first bracha of birkat hamazon (see bBrachot 48b), and that some therefore say not to say Ps 145:16 (which was written by David), and he agrees with this reasoning. Beit Yosef (siman 187, ד״ה כתב כל בו) cites this but thinks that this isn't a good reason to omit it, although Darchei Moshe notes that the custom accords with the Kol Bo.
Siddur Eizor Eliyahu (that attempts to mark the history of Ashkenazi liturgy) mentions that old Ashkenazi texts of birkat hamazon omit Ps 145:16, and that the addition was accepted by the Ḥasidim following the Ari's version, that was similar to the Sepharadi version.
On the academic side, there have been fruitful studies of early birkat hamazon versions from studying the texts from the Cairo Genizah. In particular, you might be interested in:
Shmidman's article compares the non-poetic (statutory) texts from the genizah to the poetic variants. In particular, he notes (section 2) that the while Biblical proof texts (Ps 145:16, Deut 8:10, Ps 147:2) appear consistently at the end of the poetic versions, and in an integral way, the proof texts are less integral and often absent from the statutory texts (all three are absent in about a third of the texts he surveyed). In particular (footnote 19), two-thirds of the statutory texts omit Ps 145:16, and some even continue right from כי הוא זן ומפרנס לכל right into the חתימה (doxology), neatly satisfying מעין החתימה סמוך לחתימה (the bracha must close with a text similar to the doxology). All of the above (and more — see inside) lead him to conclude that the Biblical proof texts are later additions to the statutory texts.
Ehrlich and Shmidman look more carefully at the statutory texts from the genizah for the first bracha of birkat hamazon. Although many of these do contain Ps 145:16 (see page 208, column #5), they also conclude based on the evidence (page 214) that the verse is from a relatively later layer than the core of the text.
Elizur has a different perspective, and posits that the Biblical verses are from an older layer, and compares this to the verses in the brachot around Shema, which are certainly very old. She takes issue with Shmidman's approach (page 434, footnote 47) that the statutory texts were influenced by the poetic versions, saying that it's hard to believe that the much rarer poetic texts would have such a strong influence on the common statutory texts.
Finally, to give you some examples, I quote the main representative text from each branch in Ehrlich and Shmidman's article (the last two branches have Ps 145:16, the others do not). More texts (and a critical apparatus) can be found in the articles themselves.
