I am not sure about capabilities within GEE but, this type of interpolation would be well detailed in the temporal modeling literature. This analysis could be readily be specified in R or Python using something like a spline or local polynomial regression but, the idea of daily interpolations from 2-3 monthly measurements is not statistically supported. You are talking about interpolating f(x)=365 from n=36, the confidence intervals would be far larger than the prediction margins. Within a given month, if you only had an n=2 you cannot even fit a distribution. If you did have actual daily measurements, you would find that the shape of the distribution would be notably different than a subsample of bi-weekly measurements.
I am actually not clear as to why you need daily resolution to get at vegetation phenology? Basing something like "day of the onset of growing season" from interpolated data would add a degree of uncertainty to your results that would likely be worse than random. You can pull this information from the weekly data without adding this notable error/bias. I would seriously rethink your analysis.