The Tur (552) quotes the Avi Ezri as holding that on Shabbos immediatly before 9 Av starts(meaning, when 9 Av is on Sunday) the minhag is to eat a regular seudah shlishis and then to eat a seudah hamafesket like on any other erev 9 Av. Even though, explains the Tur, one is permitted to eat meat and drink wine "afilu k'seudas Shlomo" on Shabbos before 9 Av without restriction, who says one has to eat such a seudah? In keeping with the spirit of commemorating churban habayis, one can choose to forgo indulging oneself and instead end the day with a subdued and minimal seudah hamafseket.
The Tur disagrees based on the gemara’s din (Moed Katan 2b3) that a public display of aveilus is prohibited on Shabbos. If one sets out to eat a meal designated specifically to commemorate the aveilus of churban habayis, isn’t that a public display of mourning? The Tur ends the siman by quoting the minhag of his father the Rosh to not eat a seudah hamafseket on Shabbos.
How can one explain the position of the Avi Ezri? Perhaps the A”E considered seudah hamafseket, which the minhag is to eat privately, devarim sheb’tzina, rather than public mourning, and Shabbos does not suspend private mourning.
In a more lomdish vein, perhaps one might explain the machlokes based on the chiddush of the Emek Bracha that the day of tisha b’av is marked by two overlapping themes: aveilus and ta’anis (see Rabbi Maroof’s article here). Each of these halachic themes is demarcated by similar practices of avoiding eating, drinking, wearing shoes, bathing, and having relations (see the Emek Bracha for nafka minos between the two). Within this framework, we might ask how to categorize the seudah hamafseket preceding the fast – is it part of the obligations of a ta’anis tzibur, or is it a special aspect of 9 Av mourning? If seudah hamafseket is part of the obligations of ta’anis, then just as we stop eating while it is yet still Shabbos, we perhaps may eat such a seudah to culminate our preparation for the fast on Shabbos as well. However, if the seudah is part of the obligations of aveilus, it could not be held on Shabbos.
The lomdus sounds nice, but I don’t think it fits the words. The whole point of the seudah (for those Rishonim who opine that it should be eaten even on Shabbos) is to try to demarcate our mourning in some small way even on Shabbos itself. It is for the sake of commemorating churban and not just because of the chiyuv ta'anis.