For the betas, you use the estimates from the summary() or parestimates() function from lavaan. for the parameter covariance you can use the vcov(fit) function, this will give you he variance/covariance matrix between parameter. You can subselect only the parameter you need to see the smaller matrix with the parameters you need
How many categories does your outcome has? I think you can still use this, because lavaan will give you the probit regression for categorical outcome. And the probit estimate is still an estimate of the linear relation, but linear with the ojive transformation. So, the visualization will present the linear probist relation, which can be hard to interpret.
Another option would be to estimate the simple slopes for the values of interest. And then use those estimates to transform the probit estimates to probabilities, and create the change in probablity plots on your own for each probit simple slopes. This woul be more coding work, but would lead to more meaningful plots