Sure, that's certainly fair to do. I think part of the frustration though, is that you've pointed out this inconsistency before, have had it explained to you, and then bring it up again back at square one. Continually misrepresenting the atheist position, in order for you to make it easier to argue against, is exactly what a strawman is. It is understandable if it is from ignorance or lack of familiarity, but you no longer can claim those.
But I will briefly go over your criticism one more time:
You seem to believe that if there is no absolute morality, then that means that there can be no form of morality at all. This is manifestly not the case: After all, as you say, atheists often don't recognize the existence of an absolute morality, and despite your claims, atheists by and large still behave morally. They do not act as if anything and everything were morally permissible.
Many atheists attribute the existence of morality to both evolutionary and cultural means. Just because it is believed that morality did not come from God, does not make it any less useful, practical, or ingrained to follow. There are logical, as well as nature/nurture, reasons for morality to exist and for us to maintain it.
Lastly, an atheist could believe that absolute morality does exist, but just that it does not originate from God. Perhaps the atheist believes that morality is a law of nature, written into the fabric of the universe, just like 2+2=4.