I know of a couple cases in which senators and citizens displayed battle wounds, but only under particular circumstances. Nobody walked around bare-chested daily in order to display prominent battle scars, the wearing of a toga without tunic is something that Cato used to do to publicly display his own austerity and commitment to the customs of the past (since before the tunic, a Greek garment, was introduced to the city citizen men and women wore nothing but the toga). Sallust has Marius say in defense of the accusation that his family was undistinguished, that he had gotten many scars in service to the state. But Marius doesn't actually show the scars and he doesn't just mention them alone, they're part of a whole rhetorical package:

Non possum fidei causa imagines neque triumphos aut consulatus maiorum meorum ostentare, at, si res postulet, hastas, vexillum, phaleras, alia militaria dona, praeterea cicatrices adverso corpore. Hae sunt meae imagines, haec nobilitas, non hereditate relicta, ut illa illis, sed quae ego meis plurimis laboribus et periculis quaesivi.

I cannot, to inspire faith in me, show the imagines [the likenesses of ancestors that families maintained] or triumphs or consulships of my ancestors, but if necessary I can show spears [he means the hasta pura, a military decoration], flags [vexilla were also military decorations besides being practical pieces of battlefield equipment], cavalry ornaments [phaleras, the little plates worn on the chest of a cavalry horse, also a kind of military decoration--there's some modern word that corresponds but military terminology is beyond me], other military decorations, and besides that the scars on my chest. These are my imagines, this is my nobility [nobilitas, though cognate with "nobility" refers to something very concrete, the descent from consular ancestors]; I did not obtain these by inheritance, as they did theirs, but through my many deeds and dangers

Referring to wounds got in the service of the state was not unknown as a way for Roman politicians to back up particular rhetorical points, although it's not something that everybody did every day, it occurred during very specific points. Pliny mentions a certain Marcus Sergius, praetor of 197 and great-grandfather of Catiline, who in response to his colleagues' attempt to bar him from participating in sacrifices apparently delivered a speech in which he mentioned his 23 wounds, his continued service despite these wounds, his two escapes from Hannibal after being captured, the loss of two horses from underneath him, and his famous iron right hand (for his hand had been lopped off in his second campaign) with which he continued to fight. Again, though, like Marius' speech the battle scars are only one part of the package, though Sergius' speech was apparently enough to convince his colleagues to allow him to participate in the sacrifices. I only know of a couple times when battle scars were brought up as evidence in and of themselves of the character of a man of senatorial rank, both of which include the individual's clothing being physically torn off to show his scars. Cassius Dio mentions that when Augustus exercised his censorial powers to write out Licinius Regulus from the roll of senators Regulus tore off his clothes in the curia and showed them his battle scars. Cicero provides what I think is the most interesting example of this sort of thing that I know of. During the trial of Marcus Aquillius, proconsul of Sicily in 99, in 97 the brilliant orator Marcus Antonius, acting as Aquillius' attorney, is said by Cicero to have torn off Aquillius' clothes at the end of his speech, displaying his scars and convincing the jury that to condemn a man who had given such service to the state would be an act of ingratitude on the part of the Roman people. What's interesting is that Cicero does not at all approve. Aquillius was acquitted, and in the In Verrem Cicero mentions Aquillius' trial and argued that Verres' counsel, the great (until eclipsed by Cicero following the In Verrem) orator Hortensius, was trying to use the same tactic. Cicero believed Aquillius to be guilty, and argued that the appeal to military bravery was an underhanded tactic that Antonius used because only by distracting his jury could he get his client off. Cicero urged that the same thing not happen in Verres' case:

Eadem nunc ab illis defensionis ratio viaque temptatur, idem quaeritur. Sit fur, sit sacrilegus, sit flagitiorum omnium vitiorumque princeps; at est bonus imperator, at felix et ad dubia rei publicae tempora reservandus

Now the same argument and method of defense is tried by them, and the same result is sought. He might be a thief, he might be a temple plunderer, he may be the leader of all disgrace and vice; but he is a good general, and he is lucky and must be preserved for dire times for the state.

Then Cicero goes ahead and tears to shreds Verres' claims to being a good general, but the point is that Cicero did not think that the public display and enumeration of military service and honors was at all right for a member of the senatorial class to use as a critical part of his oratory. That military gallantry was appreciated is obvious, but Cicero does not think that such an appeal rightfully has a place in such contexts. Again, this isn't something that's happening every day, which is what Cicero's upset about--it's that it's used by some people in lieu of persuasive arguments to swing political debates in their favor

There's also quite a few cases of citizens outside the senatorial class and the nobiles appealing to their military service with scars and other honors, but you've specifically asked for members of the senatorial class so I'll leave them out