The DSGE model can be written in compact form as Y_t = A Y_{t-1} + e_t, where matrix A is the companion matrix of the model. The eigenvalues here are computed based on A, which contains functions of the deep parameters of the model.

So suppose that you have a parameter issue causing BK condition to be violated, and not a timing issue, then I guess the model cannot be solved for the parameters you have chosen. If you want to keep same parameters, then you may want to modify your model slightly to change the elements of matrix A.

But first of all, check to see if it is indeed parameter issue. It is not true that changing one parameter affects ALL steady-state variables unless each steady-state variable in your model is a function of all the parameters.

And there is nothing wrong with some steady-states changing, unless you have a known target for ALL your steady-state variables.

If you have certain targets for your steady state variables, e.g., K/L = 0.5, and you don’t want that to change, you can calibrate it instead of calibrating the parameters that determines it. For example, you can declare k_l, y_l, etc as parameters and calculate (\delta=I/k), \alpha, etc based on theses. But then, you may have to solve for the steady-state in k_l (=K/L), which may be more tedious.

Remember, the steady-state is just a mapping between parameters and steady variables. Given steady-state relationship \beta = \frac{1}{1+r_{ss}}, you can fix \beta to get r_{ss} or you can fix r_{ss} to get \beta if you have a target value for r_{ss}.

But check first if your problem is indeed parameter problem.