If you don't think Nazis are critical of what they refer to as 'gender ideology' then I really don't think there's any point in arguing the toss, here.
Reading this I'm getting the strong impression that you don't understand the feminist position on gender at all.
Our objections and those who you call Nazis (by which I presume you mean the sort of Trump-voting Evangelist American Christians who are proudly right wing in the worst meaning of that word?) are not the same.
Anyway, assumptions aside, I don't think it is appropriate to use that word. It does have a defined meaning and rarely applies to the people it is used for today. I've met (actual) Neo-Nazis and they do not compare.
Anyway, no, Nazis are not critical of gender ideology in a comparable way. Half of them think that gender ideology refers to feminist liberation movements, which they oppose on principle because women are inferior in their world view. Those who could sort of vaguely be described as gender critical in the way you mean it, do so not because they oppose stereotypes but because they oppose those who do not conform to stereotypes. Neo-Nazis are macho to the extreme and despise men who are not. They also despise women who are anything other than feminine in demeanour and appearance.
The only position where we almost converge is that all males should be excluded from female-only provisions. They hold that to be true on principle, but not in practice. If they want access, they will not hesitate to claim it and the women be damned. When it comes to female-only provisions other than toilets, showers or changing rooms, they are opposed. And they agree with the former only because other men should not be allowed to lay eyes on their property. They are opposed even to refuges or rape crisis centres (violence against women and girls in that social group is high and their entitlement to control women and girls is part of their identity).
or: there are no psychological attributes of maleness because gender isn't real, and all behaviours are constructed.
And again, I don't think you understand our position. We're not saying "gender isn't real". At all. Lots of us have told you that we experience it as harmful, so do you honestly think we believe we're being harmed by something that doesn't exist?
(I will refer to gender from now on as sex stereotypes and sex role stereotypes, because that is much clearer.)
What we are saying is that sex stereotypes and sex role stereotypes are artefacts of human culture. They are human-made, and not natural, innate qualities of female and male humans.
Even though sex stereotypes and sex role stereotypes are artefacts of human culture and not something we are born with, they are imposed from birth by both our families and the society we live in. And that has consequences. It shapes the child's early development, it shapes the child's understanding of their sex and it shapes their understanding of the world around them.
That leads to certain behaviours being positively reinforced and others being negatively reinforced. The outcomes are adult personalities moulded to accept certain preferences, mannerisms, behaviours and traits as desirable for their sex and others rejected as undesirable. By which mechanism sex stereotypes and sex role stereotypes both appear true and are perpetuated.
My own belief in the feminist theory of socialisation notwithstanding, I must add that there is an ongoing, serious debate around whether some behaviours in particular are due to nature or nurture in various disciplines. There are for instance a lot of theories attempting to explain why males commit almost all violent crimes. The feminist theory of socialisation (i.e. the imposition of sex stereotypes and sex role stereotypes) is but one of them.
What the researchers who posit innate psychological attributes of maleness suggest however is not that these attributes are part of a gender identity (one's personal preferences for the stereotypes associated with one or the other sex) but part of sex differences between males and females. It remains to be seen whether they are right about that. But nothing they say supports the doctrine of gender identity in any way, because they believe as we do that socialisation plays a major role in shaping the personalities of most men and women. And that it does so along quite predictable paths.
So to recap:
Adherents to the doctrine of gender identity agree with enshrining stereotypes in policies, regulations and laws, as long as everyone can choose which stereotypes they prefer. (Your preferences dictate which sex you are.) They believe we are born with a preference for one of the straitjackets, and everyone should be allowed to wear the straitjacket of their choice, and should be allowed to try either of them on for size first.
Those you call Nazis object to the doctrine of gender identity because they believe people should always conform to the stereotypes associated with their sex. They agree with enshrining stereotypes in policies, regulations and laws, as long as that reflects traditional ideas about male and female. (Your sex dictates which stereotypes apply to you.) Deviations are undesirable and those who do not conform to the stereotypes prescribed and proscribed for their sex must be punished. They want everyone to wear the straitjacket stitched together for their sex.
We object to the doctrine of gender identity, because we reject the idea that stereotypes are natural, desirable or beneficial. (Your preferences do not dictate your sex, your sex does not dictate that any stereotypes apply to you.) We believe that sex is the only workable and fair foundation for policies, regulations and laws. We want to abolish the stereotypes so that anyone can freely live their lives according to their preferences. We want to burn the straitjacket.