I don't really know what you mean by "single case design" since this doesn't seem to be what you describe in the rest of your post. It sounds like your design would be appropriately modeled using mixed effects regression, using a random effect for subject and fixed effect for cancer stage, assuming you have multiple measures in each subject (just multiple; not necessary for it to be every stage). You'll probably need at least around 10 participants for it to make sense to use a random effect. If you don't have multiple measures in each subject then you don't need anything fancy at all, you just have a model with one fixed effect, the cancer stage. In that case this design can be modeled statistically just like if you had randomly assigned participants to a cancer stage in an experiment (if such a thing were possible, or desirable).
However, there isn't anything magic about these models and no method to have more data than you actually have. If you have few participants, your estimates will be imprecise. There is no way around this, you can't know whether differences between individuals are because of random differences between people or because of the variable you're tracking, until you have a large enough sample. Pilot studies are a poor source of effect size estimates, see for example https://www.nccih.nih.gov/grants/pilot-studies-common-uses-and-misuses
I'd be wary as well that there is some confounding between the stages you're able to observe and the unmeasured qualities of the case, that is, someone's stage 3 where you also have a stage 4 measurement is probably not directly exchangeable with someone else's stage 3 who never progresses to stage 4. I don't have a good solution to recommend for that problem, though.