I recently read this article, which appears to be one of the better ones written for how an ETI-like civilization might function, and could serve as a potential set of guide lines for the ETI.
The important takeaways are that wormhole-civilizations (hereafter referred to as ETIs) are temporally incompatible, wormholes (Pandora Gates) too far separated in time will disrupt and destroy each other, which makes it impossible for multiple ETIs to coexist in space. This gives an important use for the Exsurgent Virus.
Under this view, the ETI suspected by Firewall is simply the locally dominant ETI, and the Exsurgent Virus is the way they project their turf via STL Bracewell Probes. The Probes are intended to be found by any civilization capable of understanding how to produce wormholes (TITAN tier basically) will seek and find them, and be infected by the Exsurgent Virus. The Virus exists to coerce others into building ETI compatible wormholes which connect them, and "infect" them with the enormously complex ETI memeplex/genome, effectively causing them to simultaneously join and expand the ETI.
As the ETI is far removed from the "second singularity" of wormhole travel invention, their civilization is massively more confusing and disruptive to transhumanity than dropping the entirely EP tech base onto 1800s era earth. The absolutely insane rates of technological and economic growth, and dizzyingly crazy cultural changes (If the ETI really has 10^60 member cultures, all of which are now post-ASIs, they would be incomprehensible to lesser beings). Not to mention the basement universes of "Mature" ETI member races, and the insane logistics of inter-universal economics.
From this, we can say the Virus serves two, semi-contradictory roles:
1: To defend the ETI from damage caused by "foreign" wormhole disrupting the Pandora Gate network.
a. To disrupt emerging intelligences which are nearing the capability to create their own networks before discovering and joining the ETI.
b. To infect and disrupt the expanding colonists of other ETI-tier civilizations into their territory. An isolationist stance, but perhaps one which makes sense due to the effects of wormhole destabilization. Against ETI-tier races the Virus may simply serve as an adaptive and powerful ambassador containing the information needed to join the local ETI-Empire-Time and prevent mutual damage and post-post-singularity war between ETIs.
2: To make other civilizations join the ETI by inoculating them with the way the ETI works, and "uplift" them to post-post-singularity civilization. This is of course, at least as traumatizing and transhuman uplift programs, perhaps for so, due to the vastly greater gulf of culture and intelligence which must be bridged.
Using this variant of the ETI has elements of many of the options presented to the core book, particularly, Security, Diversity, and Enlightenment. It is compatible with Aggression Filter and War Remnants with small changes.
How this might manifest in a campaign:
1: ETI Wormhole Security: Firewall has discovered that several Titanian high-energy physics research programs have been completely destroyed via Exsurgent infection, particularly those studying Iapetus and the Pandora Gates themselves. The worst-case scenario is clear: The Virus, or its agents are specifically targeting transhumanity's attempts to understand the most powerful pieces of the TITANs (and by extension, the ETI's legacy in the solar system).
2: The galaxy consumed: Stellar measurements taken from the gates is clear: In a number of disparate locations, potentially corresponding with the age of the local gate, the stars are dimmer, and exoplanets rarer. Though likely unknown by the PCs and transhumanity, this is due to the massive rate of expansion across space-time, and the rapid and complete use of all local resources used to fuel the ETI. As the PCs/transhumanity travel closer to the "present" of ETI Empire-Time, the massive exploitation of galactic resources becomes clearer and clearer, and the universe appears more and more dead.
3: The Second Singularity: The end for Transhumanity, as promised by the Factors as the next wave of ETI colonists begin to arrive via wormhole. Is it possible to escape complete cultural change and assimilation? Should escape even be attempted? The Factors may offer assistance in the form of gate hacks which could make Sol inhospitable to ETI wormholes, or simply a place to run while chaos briefly rages. No matter what, this should probably be a campaign capstone, as the EP setting will quickly cease to mean anything as it is consumed and integrated by the incomprehensibly vast (by transhuman standards at least) ETI. Could the Factors have their own insurgent wormhole network? Are they agents of another ETI? There's a lot of things which should happen before the Second Singularity actually happens.
TLDR: A lot of unedited gushing about an inspiring article I just read. Potentially one of the more complete ideas for how the ETI may actually function.