Answer: The Gemara in Masechet Shabbat (119b) tells us that one should always set one’s table nicely (by placing a tablecloth on it and the like) on Erev Shabbat even if one only plans to eat a Kezayit (olive’s volume) of bread and one should always set one’s table nicely on Motza’ei Shabbat even if one only plans on eating a Kezayit of bread. It seems from here that eating bread (meaning washing one’s hands, reciting the Hamotzi blessing, eating bread, and reciting Birkat Hamazon) during the fourth Shabbat meal is compulsory just as it is regarding the first Shabbat meal, for both of these meals were mentioned within the same Gemara.
The Gaon of Vilna proves likewise from the juxtaposition of these two teachings in the Gemara that we can infer that the fourth Shabbat meal requires bread. The Rambam and Maran Ha’Shulchan Aruch (Chapter 300) rule likewise. Maran Ha’Chida adds in his Sefer Machazik Beracha that one should be very meticulous regarding the fourth Shabbat meal and one should have in mind while eating to “escort the Shabbat Queen out” and to retain blessing for the various meals during the week. This is based on the Gemara in Masechet Pesachim (103a) that a king is escorted out just as he is escorted in. The Sefer Tosefet Ma’aseh Rav (Chapter 39) recounts a story about the importance of eating the fourth Shabbat meal, that once Hagaon Rabbeinu Eliyahu (the Gaon) of Vilna fell ill on Motza’ei Shabbat and he vomited and was not able to partake of the fourth Shabbat meal. After falling asleep and recovering, when he awoke he asked his family members to check if dawn had broken yet, for if not, he asked for them to crumble a Kezayit of bread and spoon-feed it to him so that he may fulfill the Mitzvah of eating the fourth Shabbat meal. The saintly Rabbeinu Yosef Haim adds that partaking of the fourth Shabbat meal protects one from agonizing suffering after death some must endure in the grave.
Maran zt”l points out that according to the Geonim, although there is certainly an obligation to eat the fourth Shabbat meal, nevertheless, there is no obligation to eat bread. This is also the opinion of several of the Rishonim that one need not eat bread during the fourth Shabbat meal, and “Kisnin” bread, such as cakes or pastries (a Kezayit worth, approximately 27 grams), or fruit is sufficient.
Thus, halachically speaking, Maran zt”l concludes that it is a Mitzvah to eat bread during the fourth Shabbat meal, however, if one cannot because one is already satiated, cake or cookies are sufficient. If one cannot even eat cake, one should at least have some fruit (the words of the Chida also imply this(.