Dans l’expression foutre le camp, je dirais que c’est plutôt “foutre” qui occupe le rôle du chien : le rôle qui n’a pas beaucoup de signification à part la vulgarité qu’il apporte.

Le “camp”, c’est le campement, le matériel, d’après Expressio, et foutre aurait le sens de prendre et non de planter. Foutre le camp, ficher le camp, prendre le camp, lever le camp. J’aurais plutôt cru que c’était le champ de bataille…

I’m at a loss to answer your questions concerning foutre/sacrer (other than I think the function of chien is only technically similar to camp), but the structure of the phrase in question is similar to:
“{that} sucked/sucks the big one“ & “{that} bites/bit the big one“ (both links to The Free Dictionary),
where although “the big one” is technically the direct object of “sucked/bite” used transitively, I see it serving really more as an adverb intensifying the intransitive sense of sucked & bites when used as slang in “That sucks/That bites” to comment negatively about something:
“that [majorly] sucked/bit…[big time]….[royally],”
so maybe “le chien” could be serving to intensify (and not just complement) fucké in “Ça a fucké le chien” (oui, au moins “fucker le chien” existe d’après Wikipedia):
“Ça a [royalement/vraiment/vachement(chienement??)] fucké/foiré/merdé.”

(cf: the usage of “au maximum” and “en pagaille” following “foiré,” technically as indirect objects, but to my ear they are serving to intensify that intransitive verb, along the lines of “to the max” in English.

(Please note that although I’ve used technically bolded several times above, I must admit that I really don’t know what I mean by it here…maybe “in the strictest sense of the term”?)

Notwithstanding the above “analysis,” in an arguably half-hearted attempt to invoke the (or at least “an”) alternative, I’d offer a perhaps simpler “explanation” together with a supporting expression that I find to be perhaps more similar in structure to the phrase in question than “Foutre le camp,” to-wit:
“[Va] manger [de] la merde/marde.”

The above phrase uses a verb that usually involves pleasure together with a direct object (hereinafter “COD”) that renders it much less pleasurable (unless one’s a fly!), just as “Fucker le chien” also uses a verb for a usually pleasurable activity, but again much less so with this particular COD (unless one is also a dog or is just “in” to such things! [??not that there’s anything wrong with that??]).

Granted, the two expressions differ in that the “vulgarity” of “… manger …” is contained in the COD, while it’s the verb that contains the “vulgarity” in the expression at issue, and I apologize if this difference is relevant to your question.

Again, however, both phrases do use a “pleasurable” verb with a seemingly incongruent/inconsistent COD to create their respective “vulgar” images and impact, whereas I’ve always (^^^until just now^^^) taken the “foutre” in “foutre le camp” to mean, not its “vulgar” (to have/possess sexually) sense used with an “inconsistent” COD, but rather its “faire” sense (used congruently enough with the COD “le camp”) that was/has become/is simply a harsh and “vulgar” way of saying “leve[r] le camp” or “décampe[r],” most likely due to the word’s general use as a vulgar interjection and not to its direct association with “having/possessing sexually.”

^^^(cf [re my “until just now” above]: but then again, “faire” does not mean “défaire,” and therefore the “faire” sense of “foutre” in “foutre le camp” should perhaps not have led me so quickly to “lever le camp” or “décamper,” but rather to “faire le camp” or “camper” (sans les “dé“) which would mean the opposite of “foutre le camp.” So maybe the “foutre” in that phrase is being used in its pure “have/possess sexually” sense, which would support your claim/suspicion that the two phrases are similar enough in structure to justify further examination, e.g., is the structure of “foutre le camp” [at least when used to announce that someone him/herself is leaving a place in disgust [but not so much when ordering someone else to leave] similar/identical to the structure of the English “Fuck this place, [I’m outta here]”?)
(all that to say that I’m obviously now at an even greater loss than before regarding “foutre le camp” and its relevancy to your good question)

Étant Québécois, je dois préciser que si “fucker le chien” est une expression, c’en est une que je n’ai jamais entendue de ma vie.

Si je devais l’entendre, je présumerais que c’est juste le verbe fucker au sens de “foutre le bordel, mettre sans dessus-dessous” ou l’adjectif fucké, grosso modo “pervers, mal tourné”. Ici, l’idée étant que le chien (et on parle d’un vrai chien, ce n’est pas une métaphore!) n’est pas en très bon état après la situation dont il est question: des dommages psychologiques ou physiques en résultent. Je pencherais plus pour le psychologique: le chien a des idées ou des réactions bizarres, voire carrément dangereuses (ce qui est pas mal la même chose que quand on dit de quelqu’un qu’il est fucké).

Autrement, je peux vaguement imaginer que ce soit une variante un peu exagérée de l’expression “son chien est mort”, qui veut dire que la personne en question n’a plus aucune chance d’atteindre son but.

Si je saisis bien les commentaires ci-dessous, Celebs interviewer’aditdabenlà demandait plutôt si “chien” peut être ajouté à une expression de manière générale. Dans ce cas, la réponse est oui, mais la formulation était incorrecte: on dit en chien. Dans ce cas, c’est une formule intensive équivalente à celles qui emploient des sacres: en crisse, en tabarnak etc. C’est en fait une des nombreuses formes euphémistiques de ces expressions; en sapristi en serait aussi une.

Ça a fucké en chien signifie que la situation a vraiment, mais alors là vraiment foiré. FUBAR, comme diraient les Anglais: fucked up beyond all recognition.