First, to corroborate the staatement that antiparticles are the CPT conjugates of particles, here are some highlights from Weinberg (1995), The Quantum Theory of Fields, Volume I:
Equation (2.C.11) on page 103 in chapter 2 says that CPT converts a single-particle state to the corresponding single-anti-particle state
Page 133 in chapter 3 says, "It is CPT that provides a precise correspondence between particles and antiparticles, and in particular... tells us that stable particles and antiparticles have exactly the same mass."
Now, consider some single-particle state $|1\rangle$, and consider the state $\Theta|1\rangle$ obtained by applying a CPT transform $\Theta$. By definition, the state $\Theta|1\rangle$ is a single-antiparticle state. However, even though CPT is a symmetry, the states $|1\rangle$ and $\Theta|1\rangle$ might not behave symmetrically when the movie is played forward in time. They would behave symmetrically if the theory also had T symmetry (in which case it would also have CP symmetry); but in general, they won't.
Applying a CPT transform to a state-vector doesn't mean that we're obliged to play the movie backward in time. We can compare the state-vectors $\exp(-iHt)|\psi\rangle$ and $\exp(-iHt)\Theta|\psi\rangle$ instead, and we can do this whether $|\psi\rangle$ is a single-particle state or a state with many particles and antiparticles. This comparison gives meaning to concepts like the baryon asymmetry of the universe in the context of relativistic QFT. The point is that we don't need to know what CP means in order to talk about an asymmetry between the numbers (or behaviors) of particles and antiparticles. CPT is sufficient for this.
Sakharov's criteria for baryogenesis don't require the presence of CPT symmetry (they only require the absence of some symmetries, like CP), yet they still refer to "antiparticles." What does "antiparticle" mean if we haven't assumed the presence of any symmetry by which the particle-antiparticle correspondence would be defined? The answer might be simply that we don't need any precise QFT-based definition of "antiparticle" in this context. The criteria for baryogenesis could be expressed without using that language, something like this: If baryons do have partners with the same mass and spin but opposite charge, then they can't behave symmetrically when the movie is played forward in time, not even under a symmetry that reverses a direction of space. (And if baryons don't have such partners, then there would be nothing for baryogenesis to explain.) The idea here is that Sakharov's criteria refer to particles, not to fields. In contrast, the CPT symmetry of relativistic QFT operates on fields, with consequences for the spectrum of particles. We don't need a fields-based definition of "antiparticle" in order to understand Sakharov's criteria. The post
What are the assumptions that $C$, $P$, and $T$ must satisfy?
may be helpful in this context.