This is too long to be a comment, so I will make it an answer.
The distinction between binomial on the whole hand and Poisson and negative binomial on the other is in the nature of the data; tests are irrelevant.
There are widespread myths about the requirements for Poisson regression. Variance equal to mean is characteristic of a Poisson, but Poisson regression does not require that of the response, nor that the marginal distribution of the response be Poisson, any more than classical regression requires it to be normal (Gaussian).
Having dubious standard errors is not fatal, not least because you can get better estimates of standard errors in decent implementations of Poisson regression.
Nor does Poisson absolutely require the response to be counted. It often works well with non-negative continuous variables. For more on the underestimation (pun intended) of Poisson, see
and its references. The Stata content of that blog entry should not stop it being of interest and use to people who don't use Stata.
It's difficult to advise well on the choice between Poisson and negative binomial regression. See if Poisson regression does a good job; otherwise consider the greater complication of negative binomial regression.
I can't advise on using SPSS. It wouldn't surprise me if you needed to use other software for flexible implementation of Poisson or negative binomial regression.