I've found a nice review that has many details on plasmid replication in general, and several papers about pSC101 in detail, and I'll try to extract the key information from these papers.
First of all your plasmid as an ori region that contains so called iteron:
In many cases, the origin of replication contains directly repeated sequences, termed iterons, which are the binding sites for the plasmid-encoded Rep proteins and which have control properties. (...)iterons have been described for several replicons like P1, F, pSC101 (...).
pSC101 plasmid has three of these iterons present, along with a protein called rep that is necessary for replication. Besides these there are sequences that provide binding sites for host proteins such as DnaA and integration host factor (IHF).
Details of these can be found in this and this paper, both provides details on replication and the function of ori region in your plasmid.
As stated in this other paper, plasmid copy number is controlled by the plasmids themselves, that may vary on host type and growth conditions but:
At steady state, the 1 number of replications per plasmid copy is exactly
one per cell generation. This defines the steady state as shown in Figure 1. It should be stressed that while this is true for the population, it is not for individual plasmids-these are selected randomly for replication.
So not all plasmid copies are replicated once per cell cycle rather random ones are selected and they replicate to maintain the desired number.
When your plasmid has reached its target copy number it will adjust its replication frequency to maintain this number. This is achieved by negative control loops. There are many ways to do this (repressor proteins, anti-sense RNA) or in your case iterons. These sequence along with the rep protein may inhibit replication if too much plasmid is present by a phenomenon called “steric hindrance" or “handcuffing”. This happens because the rep protein is capable to bind two iteron, simultaneously.
At low copy numbers saturation of iteron sequences by rep proteins initiate replication, but at high copy number rep proteins tend to interact with each other and plasmids pair up through rep-rep interactions therefore cannot replicate. See figure below:
So, long story short: plasmid replication in part depends on host factors but plasmids have means to control their copy number, and once target has been reached they maintain this number in a stable form. I know this answer does not provide solution to the when part of your question, but it provides an answer to the practical part of your problem namely: plasmid numbers are quite stable after the target number has been reached. I hope this helps
EDIT: I've found this article http://www.ncbi.nlm.nih.gov/pmc/articles/PMC210920/pdf/jbacter00181-0364.pdf that says:
In all experiments, plasmids pSG21 (oriS only), F'lac (oriS oriV),pBR322,
and pSC101 were capable of replicating at all stages of the division cycle,
with no detectable differences in the patterns.
So bad news for you, because according to the paper above you cannot tell when your plasmid actually replicates. Yet still as the average copy number is quite stable due to reasons described above this may not pose a big problem. Keep in mind even if you have an extra plasmid for a short time that may still not cause problem, because transcription of your riporter gene, folding, etc takes time under which previously synthesized proteins might degrade.
However this paper says the F+ plasmid replication is cell-cycle specific, you may wan to give it a try.