How to calculate change in Loss Function w.r.t Weights for a fully connected NN
I doubt there is any ready-to-use facility but you can achieve this very easily with some custom code.
You can do -

Train your model(to the extent you want)
Get output using predict method on a dataset
Calculate the Loss using Loss function, y_true and last step output
Get the weight of every layer using Keras get_weights() API
Update weight as per your experiment using Keras set_weights()  API
Calculate the new Loss using the same process as mentioned in step#3

Keras weights API -
You can get weights as a simple array and can update accordingly.
May traverse in the loop on Layers Or access directly using array index.
Some code snippets(See reference links for detail)
Layer.get_weights()
Layer.set_weights(weights)

for layer in model.layers:
    print(layer .get_weights())

weights = model.layers[1].get_weights()[0]
biases = model.layers[1].get_weights()[1]

weights = [np.random.rand(*w.shape) for w in model.get_weights()]
# update
model.set_weights(weights)

References
Keras.IO
Stackoverflow
Stackoverflow
Stackoverflow