Is it possible to conduct a sufficiently powered factorial experiment if I have a multilevel structure to contend with?
Yes! This is possible even when cluster randomization is necessary. For more about this see Dziak, Nahum-Shani, and Collins (2012), Nahum-Shani, Dziak, & Collins (2018), and Nahum-Shani and Dziak (2018).
Note that if one or more intervention components are group-based, there is probably a multi-level structure. This is discussed in Nahum-Shani et al. (2018).
Dziak, J. J., Nahum-Shani, I., & Collins, L. M. (2012). Multilevel factorial experiments for developing behavioral interventions: Power, sample size, and resource considerations. Psychological Methods, 17(2), 153-175.
Nahum-Shani, I., Dziak, J. J., & Collins, L. M. (2018). Multilevel factorial designs with experiment-induced clustering. Psychological Methods, 23(3), 458-479.
Nahum-Shani, I., Dziak, J. J. (2018). Multilevel factorial designs in intervention development. In Collins, L. M., & Kugler, K. C. (Eds.), Optimization of behavioral, biobehavioral, and biomedical interventions: Advanced topics. New York: Springer.