Your scenario does not depend on FTL.
However, its interaction with the Doomsday Argument is more complicated and potentially weaker (assuming you accept the Doomsday Argument at all). This is because P(we live in a Kardashev ~0.85 civilisation) depends strongly in this scenario on the per-civilisation P(Doom before Kardashev 2); if the latter is importantly different from 1 (even 0.9999), then the vast majority of people still live in K2 civilisations and us being in a Kardashev ~0.85 civilisation is still very unlikely (though less unlikely than it would be in No Doom scenarios where those K2+ civilisations last X trillion years and spread further).
I’m not sure how sane it is for me to be talking about P(P(Doom)), even in this sense (and frankly my entire original argument stinks of Lovecraft, so I’m not sure how sane I am in general), but in my estimation P(P(Doom before Kardashev 2) > 0.9999) < P(FTL is possible). AI would have to be really easy to invent and co-ordination to not build it would have to be fully impossible—whether Butlerian Jihad can work or not for RL humanity, it seems like it wouldn’t need much difference in our risk curves for it to definitely happen, and while we have gotten to a point where we can build AI before we can build a Dyson Sphere, that doesn’t seem like it’s a necessary path. I can buy that P(AI Doom before Kardashev 3) could be extremely high in no-FTL worlds—that’d only require that alignment is impossible, since reaching Kardashev 3 STL takes millennia and co-ordination among chaotic beings is very hard at interstellar scales in a way it’s not within a star system. But assured doom before K2 seems very weird. And FTL doesn’t seem that unlikely to me; time travel is P = ϵ since we don’t see time travellers, but I know one proposed mechanism (quantum vacuum misbehaving upon creation of a CTC system) that might ban time travel specifically and thus break the “FTL implies time travel” implication.
It also gets weird when you start talking about the chance that a given observer will observe the Fermi Paradox or not; my intuitions might be failing me, but it seems like a lot, possibly most, of the people in the “P(Doom before K2) < 0.9999, fate of universe is STL paperclip nebulae” world would see aliens (due to K2 civilisations being able to be seen from further, and see much further—an Oort Cloud interferometer could detect 2000BC humanity anywhere in the Local Group via the Pyramids and land-use patterns, and detect 2000AD humanity even further via anomalous night-time illumination).
Note also that among “P(Doom before K2) < 0.9999, fate of universe is STL paperclip nebulae” worlds, there’s not much Outside View evidence that P(Human doom before K2) is high as opposed to low; P(random observer is Us) is not substantially affected by whether there are N or N+1 K2 civilisations the way it is by whether there are 0 or 1 such civilisations (this is what I was talking about with aliens breaking the conventional Doomsday Argument). So this would be substantially more optimistic than my proposal; the “P(Doom before K2) < 0.9999, fate of universe is STL paperclip nebulae” scenario means we get wiped out eventually, but we (and aliens) could still have astronomically-positive utility before then, as opposed to being Doomed Right Now (though we could still be Doomed Right Now for Inside View reasons).
Pardon my ignorance; I don’t actually know what SIA and SSA stand for.