I'm not sure how to tell my boss that the data is insufficient and still keep my job. Any suggestions?
Short Answer: Try your model with that data first, and if it fails after trying several times, then report the need of more resources or data. You will still have to design the model, so you can start with that and add more data as needed. Notify your boss about your worries before proceeding.
Longer Answer: The best way would be to properly design and create your model. Then proceed to evaluate and train it with your current data.
After this, you will be able to obtain some sort of correlation or effectiveness of your approach, which will probably be low due to insufficient data.
After this happens, you can approach your boss and say "Hey boss, I already built and tested the model. However, seems we got a really low performance, and I suspect it is due to insufficient training data, as I have already tried [x,y,z]."
In other words, try your model first, and if it fails due to the few data report such findings, rather than saying it won't work without trying. This way you are actually showing that you tried and this backs up your claim.
As a data analyst myself this is often the case when building models from few data. No matter how much effort and improvements you put to the model or algorithm, more data is sometimes the only way to improve your performance (specially on Deep Learning models).