Sample size computation for complex trial designs is not a bread and butter issue, as many factors must be considered, on top of, of course, the chosen alpha and beta (https://www.crcpress.com/Sample-Sizes-for-Clinical-Trials/Julious/p/book/9781584887393).
Indeed, focusing on controlled trials with repeated measures, several key assumptions on baseline effects, post-treatment effects, time-wise interactions, and treatment-wise interactions may be explicitly considered.
There are several computing approaches which can prove useful, including some R packages (e.g. https://www.r-bloggers.com/power-and-sample-size-for-repeated-measures-anova-with-r/).
My recommendation is to have good preliminary data in order to be really informed on baseline effects, time-wise changes, and intervention-wise effects. Then move on with calculations.
Otherwise, simply reducing all assumptions to a comparison between delta in control group vs. delta in experimental group (where delta is the difference between post-treatment and baseline effect in the chosen group) may lead to workable and understable computations.