Please correct me if I am wrong, but I am assuming that you mean that the calculation you want to perform is to start with a state (like the ground state), apply a local operator to that state to perturb it, and then evolve that state in time using the TEBD algorithm.
An immediate suggestion I would make is to first try to run TEBD on a finite size (but possibly large) system. If the system is perturbed only at one end, it should take time for the wavefront of the perturbation to propagate through the system, so for that time the dynamics should generally be independent of the other boundary. The time it takes for the wave to propagate will depend on details of the system you are studying such as the correlation length, and you may be able to reach sufficiently long times to see the results that you want.
If you find that using a finite system does not allow you to reach long enough time scales for your particular application, you could consider more sophisticated methods like the one discussed in the reference (https://arxiv.org/abs/1207.0652 ). In that approach, an infinite calculation is first performed to get an effectively infinite boundary condition, and then a finite TEBD calculation is performed using that infinite boundary condition. This can allow for longer time scales to be reached with a smaller finite size effect.
Please let us know if you have any questions about how to implement these suggestions in ITensor.