It's a widely-held misconception that there are any kind of "very dogmatic compositional rules" in any way in classical composition.
Maybe there once were... but I don't believe that, myself. In music theory classes, teachers enforce "rules" on student composition work (as in, "complete this four part harmony using the voice leading rules we discussed in class"). All you have to do is review a relatively small amount of actual compositions by actual composers to know that they never followed those rules. Those rules are only for theory class, not for composing. A cursory glance (or listen) at music composed from Beethoven through today will reinforce that understanding.
In fact, the tropes and genre markers of jazz are just as much "rules" of jazz composition as those of classical, pop, rock, Balinese gamelan, etc. Some things that might seem like jazz "rules" to an outsider are the common use of ii - V - I, always making it a ii7(b5) when the i is minor, playing in the Dorian mode instead of the Aeolian mode, when to use b9, #9, #11, etc. You just learn over time that doing all those little things makes music sound both more like jazz and also sound more pleasing to the ear.
Side note: I just finished a university courses in both jazz improv and theory for pop and jazz, and we had to adhere very assiduously to tons of rules in both those classes. I can't stress it enough: rules are classroom concepts, not music theory, genre, or compositional concepts.
So it is with the "rules" of SATB voice leading and counterpoint, etc. Being aware of them and using them to inform your composing will make your music sound more like the genre that the "rules" are from and usually make your music more pleasant, accessible, etc. It won't make your music sound daring or unusual, but one effective way to use the "rules" is to treat them as rules that you follow but then you choose moments to "break the rules", and those moments will stand out.
I'm just wondering if such skills can actually translate to a scope beyond classical composition
Trivially, the answer is yes. How do you think jazz came to be? Despite The Lydian Chromatic Concept, jazz music and theory was not developed in a vacuum from nothing. The ii - V - I cadence appears in many "classical" compositions of the late 19th century, to cite just one example.
Every area of musical knowledge can inform the art of composition. What will really feed into your composing voice and abilities is the study of music itself, not the music theory. However, a greater knowledge of music theory makes your study of music much more efficient.
Without knowing theory, you might read and listen to a Bach fugue and think "ok, there's some notes here.. and then similar notes starting while the first set of notes changes, and then it sounds like a third copy of the initial notes... and then... chaos?"
With an understanding of theory, you can use all kinds of shorthand: "Here's the subject... there's the answer - point of imitation is the fifth... it's a tonal answer..." etc. With knowledge of theory, you can quickly analyze, describe, and catalog what you're hearing and reading. It becomes your mental database and toolset for your own compositions. Maybe you think one day, "I'm going to write a jazz tune with two melodies that start kind of like a fugue, and the point of imitation will be a tritone". Just by having that thought, which didn't require any instrument or sheet music to write on, you have a skeleton for a new composition.
That's the power of music theory for composing.
Which music theories will help you compose? Any of them. All of them. Study taiko drumming, Cree (First Nations) songs, musique concrete, or punk rock. All of it will show you something distinct in music and you can work elements of into what you write.