Interesting question! (Which is why I took 2 days to write you an answer, of course!) The short answer is a boring “nope” though. I’m going to focus on the castrati, because I know them best, and it’s a little tidier to compare a theory of Western medicine to the old-timey Western world, and not cross both cultures and times. We have very few records of eunuchs expressing their feelings, and in the only record I know in depth (Filippo Balatri, the only castrato to write a memoir) he refers to his own body in a few ways, but not in any way I’d consider dysmorphic. It’s always tricky to read anything into historical silence, as they say, but I think I can make a decent argument that the silence here is correct: eunuchs, in societies that had a space for eunuchs, wouldn’t have had a lot of reason to feel dysmorphic about their bodies, especially compared to anyone else with a disability or mutilated body. This just is one of many places where trying to fold eunuchs into trans history will mess you up big time, because, despite it feeling kinda right to us today to put them into that trans* asterisk (and there certainly are people today who identify their gender simply as eunuch, and either live with an existing body, or if they have testes they often seek to be castrated, and they’re usually lumped in as trans people), they just weren’t trans, in the way we (“we” here meaning the cool-people-only-royal-we and not perhaps Western society as a whole) feel about trans people today.

So one of the only demonstrable ways that eunuchs expressed unique feelings about their bodies, for Italian castrati, was to refer to their bodies as diminutive, or boyish. Other people sometimes referred to them as feminine, but not castrati themselves, at least on record. Two castrati who left a lot of writing, Farinelli and Atto Melani, both in letters referred to themselves as boys, well beyond middle age. Filippo Balatri, from above, referred to himself as a “this and that” and a “little capon,” terms I would say sort of show him trying to reduce the social/mental space his body takes up. For Farinelli and Melani, their references to themselves as “boys” seems rather affectionat towards themselves, at the very least not negative, Balatri’s is maybe a bit more fraught, but Balatri was a rather unhappy person when he was writing his memoirs. But these records stretch across 3 men and 2 centuries, so they probably point to something pretty common, that eunuchs in Italian/Western culture saw themselves as more boyish, and physically “lesser” than intact men. I should probably also mention that this was usually not literally true, pre-pubescent castrated eunuchs are generally very tall, and often quite “big boned” to boot, this is amply recorded in contemporary satire and caricature, and recent disinterments of 2 castrati (Farinelli and Gasparo Pacchierotti) both estimate them, from bones, to be well over 6 feet tall. So is a 6’4” dude referring to himself as a “little boy” dysmorphic? Eh, it’s pretty weird now, but I’d argue not that weird, within his culture.

In the 17th and 18th centuries, under the theories of Humorism (blood, bile, all that good stuff) they had an understanding of sex that is subtly different from our own, which is that sex existed on a continuum from man to woman, and not a binary 1-or-0 as it later came to be thought about. The one-sex idea is argued in the book Making Sex: Body and Gender from the Greeks to Freud by Thomas Laqueur, which made a big splash in gender studies in the 90s, though it's definitely not without critics. I like it though, and I think Roger Freitas in “The Eroticism of Emasculation: Confronting the Baroque Body of the Castrato” (a “classic” paper about castrati) makes a beautiful argument for how the theory applies very well to eunuchs. I’ll summarize it here, but if you have time, register for the free-read JStor program and read it, it’s well worth your time.

Children were essentially born the same sex, and when puberty hit, boys got a burst of “vital heat” that made them hot, dry men, girls did not get this and became wet, cold women. Castrati were essentially permanent boys in their paradigm of gender, and were these liminal figures, who sort of combined humoral elements of both sexes, by being between them as, children were. This does make them “feminine,” but really not in the same way we’d see it now, with the same connotations. Filippo Balatri actually gives us a good little piece of evidence that he himself, not a doctor or even that fantastically well educated, understood his body in humoral terms:

It was established that my voice was of the finest metal, with a natural trill and good attack, outstanding fluency in passagework and a general taste for natural singing. The basis for this judgement was urgently arrived at by friends of my father’s crying out “Cut! Cut!” (and then, more than all the others, by my singing teacher) to advise my father; so that after the many cries of “Cut! Cut!” my father finally joined in too, and sent to Lucca for the surgeon Accoramboni, who was made to come to the house to stay for two months and engage me in a little gracious conversation.

This chitter-chatter was so expensive that instead of the doctoral degree (which I would have been able to obtain one day) I was given a diploma to pass as a “Cold and Injured Man” for the rest of life, renouncing forever the sweet name that I would have heard given me one day of “Papa.”

(Translation is from “Strange Births and Surprising Kin: the castrato’s tale” by Martha Feldman, published in the book Italy's Eighteenth Century: Gender and Culture in the Age of the Grand Tour.)

“Cold” here is best understandable as him referring to his body as humorally cold, as he doesn’t have testes to give him masculine heat and generative power. Pretty neat huh?

So the long answer is, they probably didn’t have dysmorphic feelings often or commonly, because eunuchs in eunuch-holding societies were very rarely isolated: they didn’t usually live without access to other eunuchs, who they could compare their bodies to, and see themselves as having a normal body, within a certain subset of normal anyway. (Balatri, notably, spent many years of his life when he was a teenager, as we all know a very tricky time to be alone, having no access to other eunuchs.) Add to this, their bodies also were understood within the framework that all bodies were understood at that time. Eunuchs were understood within humoral medicine, and any of their ailments were easily explained and treated within the theories of humorism. (Effectively or not is beside the point, it was medically normal at least.) Frankly I believe your random doc-in-a-box today has less mental space for trans people in his framework of medicine than a random back-alley bloodletter did for eunuchs. Chinese medicine, also, had (has) framework for eunuchs’ bodies. In spaces where they themselves were relatively common, their bodies were not considered terribly exotic within medicine or their own subsets of society, and they wouldn’t have had the same mental distress about their body put on them by society that (some) trans people have today.