Although there are some discussions about this point, some previous studies that published in some prestigious journals used this instrument in the context of public sector. Please look at the links below:
Are these of any help? I hope I haven't repeated the first link given above - I could not manage to access it here:
Pahi, M. H., Umrani, W. A., Hamid, K. A., & Ahmed, U. EXAMINING MULTIFACTOR LEADERSHIP QUESTIONNAIRE CONSTRUCT: A VALIDATION STUDY IN THE PUBLIC HOSPITALS OF SINDH, PAKISTAN CONTEXT.
Nawaz, M. M., & Bodla, M. A. (2010). Comparative study of full range leadership model among faculty members in public and private sector higher education institutes and universities. International Journal of Business and Management, 5(4), 208.
Mberia, A., & Midigo, R. (2016). Leadership Styles and Employee Job Satisfaction in Public Service in Kenya; Understanding the Gender Factor. International Journal of Academic Research and Reflection, 4(6), 2309-0405.
Dias, M. A. M. J., & Borges, R. S. G. E. (2017). PERFORMANCE AND LEADERSHIP STYLE: WHEN DO LEADERS AND FOLLOWERS DISAGREE?. RAM. Revista de Administração Mackenzie, 18(2), 104-129.
Hi Elizabeth, i think you should also consider which public institution you want to study. For example there is huge variation among public sector institutions in terms of leadership environment. A school leader, a hospital leader, or aa academic leader. It may be hard to find a "one size fits all" scale when it comes to public leadership. The scales used in private sector may fall short in understanding public sector leadership, because the context is very different.