First of all, you don't want an identical or nearly identical weight matrix. You won't achieve that and you don't need to. In principle a well designed model should NOT make radically different predictions when retrained, particularly with the same data, even though the weight matrices will certainly differ at least a little and possibly a lot. The same model trained two different times on the same data with the same hyperparameters will generally converge to nearly identical behaviors, right down to which types of inputs the final model struggles with. If you have the original model, original data, and original hyperparameters, definitely don't be frightened to retrain a model.

If your use case requires you to be able to strongly reason about similarity of inference, you could filter your holdout set for the inputs that both models should accurately predict, run inference for that set against both models, and prepare a small report indicating the similarity of predictions. This should ordinarily be unnecessary, but since it sounds like achieving this similarity is a point of concern, this would allow you to measure it, if for no other purpose than to assuage fears. You should likely expect SOME drift in similarity, the different versions won't be identical, so if the similarity is not as high as you like consider manually reviewing a list of inputs that the two models gave different predictions for to confirm the rate at which the difference really is undesirable.

If retraining the entire model on the complete data set is possible with nominal cost in less than a few days, do that. If not, it's worth trying transfer learning: https://www.tensorflow.org/guide/keras/transfer_learning

Note that transfer learning is a shortcut, you are almost certainly sacrificing some accuracy to avoid a prohibitive amount of retraining. You'll also still need to train the new layers against a data set that completely represents the results you want. I.E. if you train only on the new data, that's all it will know how to predict.

If you don't have the original data set, but do have abundant training resources and time, you could try a Siamese-like approach, where a suitable percentage of the training data fed to the new network is generated data with target values provided based on predictions from the current network, and the remaining data is the new data you would like the network to learn. This will probably work better when the new data is entirely novel.