If I want to understand the gender equality as an effect of GNI per capita, can FDI net inflow be an instrumental variable(IV) for GNI per capita (at PPP)?
Sorry, I can't imagine the transmission proccess from FDI to GNI. I can't imagine it as a control variable. Remember that FDI can't be properly compared with the investment of national accounts (as capital formation) ( I wrote a little on this in Spanish: https://dolarizacion.ec/2018/02/27/inversion-extranjera-vs-inversion-extranjera-directa/ ) so, it's relation with production and income is not clear.
Even if FDI is treated as a proxy of foreign participation in the local economy as a source of foreign ideas about gender equality, I'd be cautious about it: in developing economies FDI tend to be local resources that use off-shore firms for tax evasion; I think openness index ( [X+M]/GDP ) might fit better (controlling for autocorrelation, because of GDP in the index).
FDI is not a measure of capital generation for production, but a measure of foreign participation in capital. As a consequence, FDI includes financial movements that by no means increase the capital amount in the domestic economy. FDI might increase GDP (specially if it is greenfield investment) but it might not (specially if it is merger and acquisitions), FDI statistics usually aren't disclosed enough to say. There are several papers with empirical data about it.
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In the case of GNI (which relation with GDP is also disputable), I find difficult to use it as an instrumental variable. I'm just warning.