The usage of SPQR has been discussed before, for instance here by u/A_Sheep_From_NZ, u/Sansa_Culotte_ (most relevant to your question), and others, here by u/XenophonTheAthenian and others, and here by u/J-Force. It seems that the emperors valued the legitimacy of tradition more than being personally mentioned in this case. And it is not like the emperor was not already present in the symbols and the minds of the military. In fact, when it comes to adding something to the abbreviation, Pliny the Elder (N.H. 33.7) claims that the Ordo Equester (sometimes translated as knights) was added after Populus as a recognition of its role as the third corpus of the Roman state (I do not think this has been verified archaeologically, but it is an interesting claim, especially since Pliny himself belonged to that order). The late source Historia Augusta (Commodus 12.7) also states that Emperor Commodus once mentioned the army before the Senate in a proclamation, which has been interpreted as a slight against the latter. So it seems that no emperor was interested in adding themselves to the traditional SPQR, that other elements of state were more relevant to add and that there were other ways to decrease the Senate's perceived importance

Edit: I should add a source for the interpretation of Commodus' proclamation, this was interpreted as such in the article "Commodus the God-Emperor and the Army" by Michael P. Speidel