In battle, there was no expectation that a defeated general should deliberately seek out his own death in the Late Republic (Rosenstein 120). However, defeated generals often did seek out death if the alternative was to be captured, and a death fighting in battle was considered somewhat glorious even in defeat (120-2). The greater the disaster, the more advantageous death or suicide could be to one's reputation (122-3). The fact that deliberate seeking death in battle could somewhat remove culpability from a general who had done everything he could but had been defeated is well demonstrated in this excerpt from Caesar:

"When all were panic-stricken and Curio understood that neither his exhortations nor his entreaties were listened to, considering that in such pitiable plight only one hope of safety remained, he ordered them in a body to occupy the nearest hills and the colours to be transferred thither. These, too, were outstripped by the cavalry sent by Saburra. Then indeed our men touch the extremity of despair, and some are slain as they fly from the cavalry, others fall to the ground unwounded. Gn. Domitius, prefect of the horse, surrounding Curio with a few horsemen, begs him to seek safety in flight and hurry to the camp, promising not to leave him. But Curio declares that he will never present himself again before the eyes of Caesar after losing the army that he has received from him on trust, and so dies fighting. " (Caes. Bell. Civ. 2.42).

The example of Caepio after the catastrophe of Arausio shows that it was most important that a general demonstrated some personal courage in battle, even a defeat, even if his strategy and tactics were responsible for the defeat. Varro seems to have avoided such charges after Cannae because he fought, and he re-gathered as many troops as he could after the battle and continued campaigning (though fairly passively) with his consular army. Caepio however was convicted of treason after Arausio for his personal cowardice rather than his negligence that led to the defeat in the first place (123-4). Actual tactical competence was not the main qualification for military leadership in the Republic, but rather superior moral character, especially virtus. A failure to demonstrate virtus, rather than a failure to win a battle, was the main crime a general could be accused of and was harshly punished by one's fellow elite, unlike for example corruption, because it brought into doubt the entire basis of moral authority upon which the senatorial class claimed the right to lead (151-2). As such even defeated generals who had a reputation for virtus, that is personal courage, could enjoy successful careers (152).

Still in the military context but perhaps closer to the crux of your question, there was no peculiarly Roman association between shame and suicide. Rather than being associated with shame or pudor, suicide is usually associated with desperation, or desperatus salus (Rauh 407). Unlike other cultures some might mention like that of Japanese bushi in some periods, pudor or shame does not appear to have been "a strong force for guiding behavior" in this regard. Note for instance that "there are no known instances of politicians committing suicide after expulsion from the senate" (407). As shame was not a key factor in suicides for Roman elite, the suicides that did occur we can not "exhibited little capacity either to redeem the agent's honor or to ennoble him" (407-8). Some depictions of battlefield deaths such as that of Paulus at Cannae or P. Crassus (the son of the general) at Carrhae carry an element of redemption in the literature, but these are fairly isolated examples according to Rauh (408). Rauh (as does Rosenstein) argues that while battlefield deaths could be redemptive for one's reputation, battlefield suicides were usually desperate measures "that often evoked pity or compassion rather than inspiration or respect". This trend can be observed in the traditions of soldier suicides at Trasimene, Casilinum and Carrhae (408). Suicides in foreign wars were therefore more of an attempt to avoid capture and humiliation rather than an effort to redeem one's reputation, and the case of Varus at Teutoberg in AD 9 is an ideal example of this tendancy.

Importantly, suicides among the Roman elite were most prominent in periods of civil war, and we can note that even in this period wartime suicide remained rare in foreign wars. According to Rauh, in addition to the increased rate of suicide, "emblemenatic cases such as Cato and Brutus changed the cultural resonance of suicide" (408). Interestingly, in this context we might conclude that suicide was a different phenomenon in the context of civil conflict, where factors such as honour and shame and specifically concern for one's position in a turbulent society did become relevant factors in the decision to commit suicide, whereas suicide in foreign war was a rare phenomenon and rarely a response to concerns of honour and shame (409). Note that the excerpt from Caesar I provided above is in the civil war context, and Curio emphasises his shame upon disappointing Caesar by losing the army entrusted to him in explaining his decision.

As such, suicide by elite Romans was a different phenomenon depending on the context in which it occured. In wars against foreign enemies, suicide was usually a product of desperate circumstances and especially a product of total despair or a desire to avoid capture and torture or humiliation, as in the case of Varus at Teutoberg (400). Honour or shame were not guiding factors in driving generals to suicide in such instances. However, suicide was far more common in civil wars, which during the Late Republic were very prevalent, and the intra-cultural nature of these conflicts appears to have placed a different and more intense emphasis on maintaining honour, avoiding shame, and using suicide as a means of coping with the latter. As such, if it was ever true to say there was a culture of "honour suicide" in Republican society, it was during the civil wars of the Late Republic. There does not appear to have been any particular etiquette associated with it, other than dying in battle or killing oneself with a manly weapon such as a sword appearing to be desirable. This is not to say that Rome's "political and military elite" naturally had a culture of honour suicide in this period, as they continued to kill themselves rarely in foreign wars, but rather that the civil wars themselves fostered a culture of honour/shame-related suicide.

Sources:

Caesar, Bell. Civ., Loeb translation.

Rauh, Stanley H. “The Tradition of Suicide in Rome's Foreign Wars.” TAPA 145, no. 2 (2015): 383-410

Rosenstein, Nathan. Imperatores Victi: Military Defeat and Aristocratic Competition in the Middle and Late Republic. Berkeley: University of California Press, 1990