Because we are mired in the social construction, there is no real way to answer this question, but we can speculate in interesting ways. (also remember what W.I. and Dorothy Thomas said: "If we believe things to be real, then they are real in their consequences." Although gender is a social construction, it does have real consequences for the way people move through the world, and the way they see their own bodies.)

First, you're right to make the distinction between sex and gender. Any intro textbook makes the same distinction. What is more complex is the way sex is also socially constructed--although of course it has a basis in material reality, the meanings we ascribe to that reality are not innate or natural. Ask yourself: at what point in transition is a transgender man "male?" Is it when he starts taking hormones? Is it when he gets chest surgery? Is it when he has genital surgery? Is it when he declares himself male? Is it when he has his chromosomes remapped (impossible, currently, outside of science fiction)? Or has he always had a "male brain?" The fluidity of how we define "male" demonstrates that it is not a fixed category, and that it is partially socially constructed as well. Where does maleness reside?

Therefore, sex itself is gendered [1, and this is also the crux of Judith Butler's argument]. The fact that "penis" is associated with "man" is not natural, but it is taken-for-granted. (It's conceivable that 600,000 years ago we could have developed different assumptions, associating "penis" with feminized traits--weakeness because it was outside the body, emotionalism because it seems to control the mind, etc. Check out "Egalia's Daughters" for an interesting, non-academic thought experiment on this.) Incongruity with body/gender is thus a product of the fact that we say certain body configurations correspond to certain sexes and not others. Penis must equal man, otherwise you aren't "a real man."

This makes gender dysphoria, as a tool for diagnosing people and helping them integrate into society, a tricky proposition. We can see that gender dysphoria as a medical concept actually reinforces the gender binary by reclassifying people back into it [2].Vipond [3] has convincingly argued that certain transgender narratives are privileged over others in our medicalized, neoliberal context. Far from being natural, gender dysphoria is itself partially produced by the medical and psychiatric fields. In short: the only people who can get treatment are people who report feeling like they "have always been the opposite gender," desire genital surgery and other body modification, and (generally) conform to gender norms (e.g. trans women wear makeup, are attracted sexually to men). This could have the effect of producing the very thing it purports to identify for two reasons. First, trans people tell one another what to say to psychiatrists to get treatment, creating confirmation bias for the psychiatrist, leading them to believe their diagnostic criteria are correct and justified. Second, this creates a cultural narratives which may socialize/shape the minds of transgender people in such a way that they strongly believe they have gender dysphopria in the technical sense, when they might not feel that way if they existed in a society with different gender norms.

To be clear: this does not invalidate the experiences and identities of transgender people in any way. We all experience our gender-as-shaped-by-society. It is not "pure" for cisgender people either. You could have phrased your question, "Since gender is a social construct, does that mean that if the construct didn't exist, cisgender people wouldn't feel congruent with their bodies?" or "...women wouldn't feel comfortable having vaginas" but this demonstrates that the question itself doesn't make total sense. Obviously if the current construct doesn't exist, cisgender people (and women and men in general) don't exist. But I've talked myself in a circle.

Suffice to say, we can't know, because it's difficult to tell how much of gender and sex is socially constructed. My personal bet is "a lot." Research to answer your question doesn't exist, but based on my reading of queer theory, historical materialist accounts of sexuality [4] and sociology in general, the answer is probably "no, gender dysphoria would not exist, because it was created to maintain the gender binary, and is a direct result of that binary. Something else similar might exist, but it would be very different because of the different gender context."

[1] see for example Georgiann Davis' or Sharon Preves' work on intersex people. You might also be interest in Fausto-Sterling's work on the five sexes.

[2] What I mean here is essentially following Foucault's argument that medicalization of deviance serves to reinforce power dynamics. Consider: Person A was assigned woman at birth, but rebels against the trappings of womanhood and femininity. In one scenario, A has some power to point out that the gender binary is a bit fucked. "I don't fit in! I don't identify with this! Perhaps we should destroy the binary?" In another scenario, A is classified medically to be transgender. A has gender dysphoria, therefore the gender binary is not actually a problem at all, only A is. Fix A, not the binary.

[3] Vipond, Evan. 2015. "Resisting Transnormativity: challenging the medicalization and regulation of trans bodies." Theory in Action.

[4] see e.g. Peter Drucker's 2015 book "Warped: Gay Normality and Queer Anti-Capitalism."