to continue even when an infeasible solution is found during an SDDP system optimization.
No. A fundamental assumption is that your system has relatively complete recourse and that it can never end up in an infeasible state.
Frequently running into problems where a primal solution is infeasible
This might be due to your model, but I find that Clp is unreliable and also reports feasible problems as infeasible after a number of iterations. Use Gurobi instead.
Read: https://odow.github.io/SDDP.jl/stable/tutorial/basic/06_warnings/