How to estimate a constant elasticity of substitution for a country in a solow model including human capital and scarce resources
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You might want to have a look at MRW (1992) https://academic.oup.com/qje/article-abstract/107/2/407/1838296/A-Contribution-to-the-Empirics-of-Economic-Growth and the literature on growth regressions in the handbook of economic growth and the handbook of macroeconomics. In addition, Acemoglu's textbook on economic growth will provide you with some useful references as well as a clear and concise introduction to the topic. Although, I believe that you'll also be a little bit disappointed as you'll find that something close to what you wish to do has already been done.