What you are doing is the three-way interaction version of saying a significant simple effect and non-significant simple effect equals a significant interaction -- which is false. However, there is that famous saying that God surely loves the p=.06 as much as the p=.05. Really, though, what it sounds like is your hypothesis is slightly off, or at least what you thought your hypothesis was is slightly off.
Even if you could get your three-way interaction significant, if I were a reviewer on the paper (or a thesis advisor, etc.), I would check to make sure the pattern of results fits your hypotheses. It sounds like you expected the AB interaction component at C1 to go one way (let's say positively) and the AB interaction component at C2 to go the other way (let's say negatively). Even if there is a significant 3-way interaction, if your hypotheses really predict 4 simple effects of A that form a three-way interaction (e.g. significantly positive at B1C1, negative at B1C2, negative at B2C1, and positive at B2C2), then just your three-way interaction doesn't fully support your hypotheses. You would need a significant 3-way interaction, 2 significant simple interaction components (AB at C1, AB at C2), and then 4 significant simple effects in the predicted direction (A at B1C1, etc.).
All this is really to say, do your results mean you need to retool your hypotheses, and then run a replication that is properly powered to find your new, better hypotheses? If your hypotheses predict a significant interaction at C1 and then an attenuated effect at C2, then your data already does qualitatively fit the hypotheses, and the question is whether p=.06 is OK to report in a paper in your field (also you should be checking the simple effect of A at B1C1 and B2C1 for the significant AB interaction at C1). If it is not, then replication is still needed, and be more conservative with your expected effect size.