He may develop dysmorphia... but probably not for the reasons you think
Ultimately it is a choice as an author if you want a specific character to have dysmorphia one way or another; so, instead of answering IF he will develop it, I will just answer the actual WorldBuilding aspect of the question which is to identify the risk factors involved in if your character is likely to develop dysmorphia based on the biological and psychological principles at play here.
How humans develop the concept of gender identity
Children tend to develop a strong sense of gender classification around the age of 3 or 4. At this age, children begin to decide what makes a man or a woman based entirely on what they see modeled most often. You can be a large, well muscled, deep voiced, full bearded man, etc... but if you walk up to a 3 year old with long hair, a kilt, and pink polo shirt many children will have a very hard time figuring out if you are a man or a woman, because young children believe that fashion is just as much an indicator of gender as less voluntary expressions like breasts or facial hair.
However, people don't naturally learn that a beard is a better predictor of gender than fashion choices. What they learn is that they find themselves mistaken more often if they judge a person by something like clothing; so, they create a mental hierarchy of what makes a man or a woman, but these hierarchies are purely based on experience. No teanager or adult is walking around with a DNA kit to see if you have an X or Y chromosome, and they hopefully aren't pulling your pants down to see if you have a penis or vulva. All most of us do is continue to refine those lessons we learned as kids and apply them dozens of times a day to assume other people's genders. We start off looking for things that are usually strong indicators like secondary sexual features because they are the most reliable, then move on to cultural norms if the secondary sexual features are inconclusive.
How Aromatase Excess Syndrome affects gender identity
The thing about Aromatase Excess Syndrome is that is does not present itself until late childhood: well after a child has established their gender identity, and their hierarchy of gender determinators. So, if the child grows up with a penis for the first 12-15 years of their life, and suddenly, they get boobs, then the most likely outcome is that they will see themselves as a boy with boobs, because they are already past the age where they have decided that they are a boy.
Infact, most cases of gender dysmorphia have nothing to do with hormones or what the person looks like when they become a teenager. Recent research shows that the vast majority of cases of gender dysphoria can be traced to early childhood trauma. Feelings of self-hatred and self-rejection are common trauma responses; so, when you experience trauma at a young age when you are still developing your idea of "what is gender", that rejection can easily be targeted at one's own gender causing a trans identity later in life.
Since children with AES are indiscernible from children without it during thier formative years, there is no elevated expectation of trauma, and therefore, no elevated expectation of dysmorphia. If they opt for any "corrective" surgery or hormone therapy later in life, it will in most cases be to reinforce thier male identity unless they experience very powerful adolescent trauma directly related to thier male identity. It's also impossible to say how often men opt for corrective treatments because of gender issues because there are underlying health benefit in correcting a person's hormonal balance that are being considered as well, even when the man is perfectly fine with how he looks, he may want treatment to reduce the risk of cancer or to fix brittle bone issues.
Simply being bioengineered is a much greater risk factor for gender dysmorphia
A poor parental connection is the most common source of early childhood abuse. It is such a significant determining factor that foster children are 15 times as likely to grow up to identify as trans as the general population. Since a bio-engineered child has no biological parents, ther formative years will most likely resemble that of a typical foster childhood; so, the development of female traits in adolescence may come as a welcome relief from the sense of self-hated that they are much more likely to already have. So, the AES will not make them more likely to have dysmorphia, but it may make it a lot easier to adopt a trans identify when they get there.
So, indirectly, being bioengineered may cause a man to be more welcoming of an AES body, and less likely to seek out corrective therapies.
But, being a future tech civilization should also reduce the risk of gender dysmorphia
As the connection between Trauma and Gender Dysmorphia becomes better understood, therapists have been getting a LOT better at treating Gender Dysmorphia. Back in the 1950s-1970s therapists attempted to treat Gender Dysmorphia using shaming techniques which made the Dysmorphia even worse. These early failures in therapy lead to the recent belief that it is only treatable through Gender Affirming Surgery; however, now that it is understood as a trauma response, therapists are beginning to transition to using trauma therapy techniques like NARM and DBT which are proving to be both way more effective and safe for patients than Gender Reassignment. When you address the trauma that causes Dysmorphia, the Dysmorphia itself disappears.
As time goes on, the expectation is that the availability of proper therapies for Gender Dysmorphia will become more prevalent and effective. So, even if bioengineered children are more likely to develop Dysmorphia, they will also be more likely to have access to the therapy services that they need to overcome it.
When it comes to accepting thier adolescent bodies, better trauma therapies could make your bioengineered men more likely to peruse surgery for medical reasons, but less likely to care what thier bodies look like if they have the skill set to deal with shame in a healthy way.
