Chris:
Your problem is very similar to the toy problem in the paper ", “Sparsity information and regularization in the horseshoe and other shrinkage priors”, Juho Piironen and Aki Vehtari. The toy problem allows for multiple signal measures, whereas your problem allows for only a single signal.
I have written up the Rebar approach to solving this problem and it is described here. Finally A Way to Model Discrete Parameters in Stan
You can try both approaches, Rebar and Horshoe. I would be interested to find out which performs better on your actual data.