Using poisson family for log+1 counts in glm

Using poisson family for log+1 counts in glm

I'm running a GLM using counts as my response variable and would normally
use family=poisson. Yet I'm also running SAR models, which couldn't easily
handle my counts as there was such a small amount of variation between
them (high number of counts at very low numbers) so my supervisor
suggested I use the hack of log(counts)+1 as my new response variable.
Should I be using the family poisson?
The use of the family poisson will give more conservative results
(significance wise) when compared to the same model without poisson family
specified, but will obviously give me warnings as I have non integers.
Because of this (I assume) it wont give me an AIC either.
Any advice would be awesome!