Synthetic data generation is a hot and new topic at the moment. Im writing my MSc thesis on time-series synthetic data generation using TimeGAN.
First, overfitting is a general problem, which can also occur if you have lots of data. Having little data is not necessarily an indication that you will therefore have overfitting. If your model overfits, you will have to reduce your model complexity or add regularization methods. This is both with your predictive model as it is with your generative model. Whatever you decide on, you will need to deal with overfitting models, either with the generative or the predictive model.
Second, you can indeed generate more data using synthetic data generation methods. But your generative model will only be able to learn to generate data similar to the data in your training set. So if your training data does not cover the complete distribution of possible data points, your generative model will not suddenly start generating those.
I think you can fit a small time-gan model to your data effectively without it necessarily overfitting. However, your additional data is not necessarily going to provide your predictive model with more 'information' as the newly generated samples are based on the original points. Implementing TimeGAN is a lot of work and tuning it is hard. I reckon you can better try to fit the predictive model without generating more synthetic data and spend your time optimizing the predictive model instead of implementing and tuning TimeGAN.
Hope this helps