Poisson would actually work as well as its log probability function can evaluate on continues data.
I think you would want to model it with something that could be parameterized as location + scale, which you can model the change in average spending or standard deviation of the spending. With that in mind, you can try specifying the likelihood with Gamma, log-Normal, Truncated Normal (not half Cauchy or half Normal as the location parameter is set at 0).
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