Yeah, if your design matrix is that poorly conditioned then you have bigger problems. And adding noise to the diagonal isn’t going to help.
In a very real sense this isn’t even a statistical problem but a computational one – floating point arithmetic only has so much accuracy and if you’re trying to build a model using data with such extreme scales then you’re never going to get around this limitation.
Standardize your covariates and perhaps even your variates. Or, even better, figure out the appropriate units for each covariate and use those so that everything is O(1).