This boils down to timeseries forecasting which is particularly challenging for financial data because finance exhibits all the things that makes forecasting hard: non stationary, low signal to noise, etc.
In the case of forecasting deposits, I very much doubt that you will be able to get satisfactory results even if you do manage to figure out all the relevant variables. And then you will have a problem called curse of dimensionality because you will never have millions or thousands (or even hundreds) of years of data to train your model.
One thing to be aware about machine learning algorithms in general is that they are (can be) very good at interpolating and pretty bad at extrapolating, which means your model will fail when you need it the most.
So my answer to your question "Do you see a problem with this approach?" is that you might get ok forecasts when is doesn't matter, ie, when deposits are not really changing, but forecasts may fail miserably when you are in unchartered territories.
And this goes for forecasting just a few months in advance... never mind three years.
Regarding the second question, I would try with and without core deposits to see if there is any explanatory power.
A useful starting reference would be tensorflow's tutorial on Time series forecasting which uses Convolutional and Recurrent Neural Networks to forecast weather. You could try to use it with your data to forecast out of sample periods.