I am trying to model that death counts follow a Poisson distribution, i.e, D(t,x)~ Poisson( E(t,x)*m(t,x) ), where D(t,x) is the number of death at age x in year t, E(t,x) is the exposure to death at age x in year t and m(t,x) is the mortality rate at age x in year t.

Now, from the website, the raw death counts, D(t,x), are not available. Instead, the death numbers are adjusted by Lexis triangles. Thus, the death counts are non-integers. However, Poisson is for non-negative integers. What should I do?

This is just a suggestion (not from experience). Have you considered using the cumulative Poisson distribution? I.e. P(a<X<b) where a,b are the first integers on either side of the reported value. I also found this paper which made use of Lexis diagrams in constructing cohort models which may help.

Counts adjusted by Lexis triangles are a count multiplied by an estimate of the proportion of the cohort in the given age category. A fairly complete approach would be to work with three sub-models: 1) model the unobserved counts; 2) model the aging process that generates the proportions; and 3) model the observation of the counts… or if the counts are large and you don’t care about the details just round them and use the Poisson.