I feel that I understand what you are trying to solve but there are many « loose ends » in your statement of problem that it is difficult to be affirmative. So, this is not really an answer, but rather a collection of observed « loose ends » and some discussions on how to get rid of them.
- « orbit » vs « trajectory »
It is not clear whether, for you, they are synonyms. Take this quote from your question
SV1 at epoch T1 that will result in orbit 1 (let's say the trajectory propagated for 1 period)
This can be interpreted that orbit is a whole, while « trajectory propagated for 1 period » is just a part of that orbit.
Alternatively (and I believe this the right approach), we can say that the term « orbit » conveys the notion of something moving (hence, having a trajectory), but remaining in the vicinity of something else. Therefore, an orbit is a trajectory, but not all trajectories are orbits. I think, for example, we can’t talk about the "Voyagers’ orbits", at least not any more (I am eager to know if somebody can make a rejoinder).
- Can « orbit » be defined as a closed trajectory ?
An object has a closed trajectory when it returns to the same positions after some time. Do orbits need to be closed trajectories ? Real life tells us there is no such « closed orbits », or at least, they are extremely rare. If the orbital plane changes with time (and it has to because of Earth oblateness, among other things), it is difficult to have systematic closure.
- What is exactly the « period » of an orbit ?
The same quote above from your question seems to imply that a state vector (position, velocity) at a given epoch determines a closed trajectory in some judicious reference, so that you can define a period as the minimum time needed for an object to come back to its initial position. Hence, the rest of the trajectory is just a repeat of this part, ad-infinitum.
While this notion is correct in a 2-body situation of orbital-mechanic theory, in a judiciously chosen reference frame, it is easy to observe that it breaks down when there are other forces and also when the reference frame is changed.
You may get away with this difficulty by restating that : SV1 at epoch T1 defines a Keplerian orbit, with period P, in an ECI reference. The actual trajectory between T1 and T1+P and the Keplerian orbit are the objects of interest (but distinct).
- This is my guess of what you are trying to formulate.
Consider an object orbiting a celestial body and its actual trajectory. If we are given exactly a series of state-vectors SV1,SV2,SV3, … at sampling instant T1,T2=T1+P,T3=T2+P, … can we use the Keplerian orbits, which are determined uniquely by SV1, SV2, SV3, … to approximate the actual trajectory, to within a certain tolerance and within a certain finite time duration ?
P needs not to be exactly the Kepler period and can be used to control the approximation errors.