I think the ctsem package implements some dynamic linear models with Stan, so that could be some inspiration (although not exactly a case study), also I am not sure they actually use gaussian_dlm_obs…
Also the Chronikis package Chronikis: a language for Bayesian time-series models that compiles to Stan by @Kevin_Van_Horn could be useful (Kevin might also have some worked examples to share)
Best of luck with your model!