If I have a generalized linear model (GLM) with a particular likelihood, and I have another GLM of the same data (say nested within the first model), I can compare the model performance using Akaike information criterion (AIC) and say which model is better.

Let's say that I fit a Gaussian GLM, so the usual OLS linear regression.

However, I know that my $Y$ is always an integer, so I wonder if I might get a better fit by using an integer-only likelihood, and I fit a Poisson GLM.

Assuming the same features, what, if any, meaning is there to the difference in AIC values between the Gaussian model and the Poisson model? That is:

$$ \mathbb E[Y\vert X] = \beta_0 +\beta_1X \implies AIC_1\text{, based on Gaussian likelihood}\\ \log(\mathbb E[Y\vert X]) = \beta_0 +\beta_1X \implies AIC_2\text{, based on Poisson likelihood} $$

Is there any meaning to comparing $AIC_1$ and $AIC_2?$ The answer here seems to indicate that it is not meaningful to compare $AIC_1$ and $AIC_2$, but the comments express some dissent, and the answer here also seems to indicate that it is not meaningful to compare $AIC_1$ and $AIC_2$, but the measure theory is a bit much for me.

(Then again, Kjetil's, "If one contemplates both continuous and discrete models, these two kinds of models cannot be compared with AIC, since they use different dominating measures (Lebesgue measure, counting measure)," seems pretty definitive.)