I've been estimating continuous positive outcome Poisson regressions with the Huber/White/Sandwich linearized estimator of variance fairly frequently. However, that's not a particularly good reason to do anything, so here are some actual references.
From the theory side, $y$ does not need to be an integer for for the estimator based on the Poisson likelihood function to be consistent. The data does not even need to be Poisson. This is shown in Gourieroux, Monfort and Trognon (1984). I believed thisThis is called Poisson PMLE or QMLE, for Pseudo/Quasi Maximum Likelihood.
There's also some encouraging simulation evidence from Santos Silva and Tenreyro (2006), where the Poisson comes in best-in-show. It also does well in a simulation with lots of zeros in the outcome. You can also easily do your own simulation to convince yourself that this works in your snowflake case.
Finally, you can also use a GLM with a log link function and Poisson family. This yields identical results and placates the count-data-only knee jerk reactions.
References Without Ungated Links:
Gourieroux, C., A. Monfort and A. Trognon (1984). “Pseudo Maximum Likelihood Methods: Applications to Poisson Models,” Econometrica, 52, 701-720.