Can I conduct an exploratory cross-over study on data originally collected in a clinical trial?

I need your opinion about this study. Originally it is a clinical trial comparing two treatment regimens (of IVF) on the chance of fertility. One (Treatment A) consists of ovarian stimulation during two follicular phases of two consecutive ovarian cycles. The other (Treatment B) consists of two ovarian stimulations on the same cycle (one follicular phase, the other luteal phase). Now the investigators want to use these same data to do a study in the Treatment B group. To compare the hormonal parameters (and other numerical or qualitative parameters) following each treatment B stimulation. So the participants remain their own controls. We have N1 the number of patients treated (stimulated) during cycle 1 (follicular) and N2 the number of patients treated during cycle 2 (luteal phase). Not all women make it to cycle 2 (some get pregnant before, some are lost to follow-up and some for other reasons)

What might be the most appropriate method here? Knowing that the study is not designed for a cross-over, no wash-out period. The question of the carry-over effect that arises (in principle, stimulation 1 should not have an impact on the outcomes in cycle 2, at least that would be the hypothesis). In any case, I think that we can do an exploratory study.

If a crossover is the most appropriate, should we impute for women who did not go to the second phase (in ITT)? and analyze per protocol only women who did both cycles? Are there other approaches to take into account all women in phase 1 and all women in phase 2 without imputation?

Can we do analyses without matching? Thank you for your help.

Translated with (free version)