Methods that can be used to tackle endogeneity problems include two-stage least squares and instrumental variables. Also, the mean level of the dependent variable should be identical across groups before the experiment is run - before units are assigned to the treatment variable versus the control group :)

You have a relatively long T and N, GMM may not be the appropriate due to the possibility of instrument proliferation. You can use other methods like panel ardl or cross sectional ardl variants. CUP-FM can also work in here as it controls for endogeneity and it is T ans N asymptotic

First, check if endogeneity exists in the data; then, proceed to 2SLS or GMM with an instrumental variable approach.