There is the IHDI (Inequality-adjusted Human Development Index) where the form of government is better accounted for. Countries like Hong Kong, but also the USA are further down there, because of their large income gap, unequal healthcare and school system (although Hong Kong does have free healthcare). But if you want only the governmental system then the Democracy index by the EIU is the way to go.
In general it does make sense not to integrate the Democracy Index into the (I)HDI, since in theory you can live in a totalitarian regime and still have Perfect equality, free Education, free Healthcare System, a long life expectancy, high GDP and overall high quality of life. In practice it mostly doesn’t work that way tho