din.org.il was posed a related question (in Hebrew) titled: "Kiddush on Shabbos on Pas Haba B'kisnin"
Greetings.
In a pressing situation if there isn't wine or bread, is one allowed to make kiddush on pas ha ba b'kisnin?
בדיעבד אפשר לקדש גם על פת הבאה בכיסנין – עוגה, ואף אם לא יקבע עליה סעודה נראה שאינו מעכב.
B'dieved one could make kiddush on Pas Haba B'kisnin- cake, and even if one doesn't isn't kovea a meal on it it would appear that it doesn't prevent [one from fulfilling their obligation]
The sources cited for this ruling are:
שמירת שבת כהלכתה פרק נג סעיף ז, ומקורו בשו”ת יביע אומר ח”ג סי’ יט ושו”ת רבבות אפרים ח”א סי’ רא, וראה שש”כ שם סעי’ יח שטוב לקבוע סעודתו על העוגה באופן זה, אבל אינו מעכב.
Shmirat Shabbat K'Hilchasa perek 53 si'if 7, Shu"t Yabia Omer cheilek 3 siman 19, Shu"t Rivivos Ephraim cheilek 1 siman 201, and see there written in siman 18 that "it's good to kovea one's meal in this manner and it doesn't hold one back
Application of this ruling to Lechem Mishnah
One could perhaps extend din.org.il's ruling to your question.
If one is able to fulfill their Kiddush obligation-- which many believe is on a d'oraisa level (Rambam, Hilchos Shabbos 29:1, see also Mishnah Berurah 271:2)-- via the use of pas haba b'kisnin,
all the more so the obligation for Lechem Mishnah-- which virtually everyone says is d'rabanan (besides the Taz OC 678:2 -- but even his opinion can mean d'rabanan- see here)-- can be fulfilled with pas haba b'kisnin.