If you’re modeling the process as a linear combination of normal distributions, can’t you just normalize your data so that it actually gives the density of emissions, and estimate the five parameters giving the coefficients of this linear combination (with the constraint that they sum to 1)?
Maybe that could be formulated as a multivariate normal with a diagonal covariance matrix, since that will be normalized to unity without additional constraints - and the \mathbf{\mu} vector components would give you the relative intensities, I’m not sure about that, but check, maybe.