As thick as it is, and the speeds involved in conventional rocket launch, it would have some pretty serious drag, and some pretty serious heat build-up as well.
Sorry, doesn't really work that way... the fin is very thin in "front profile"... far more so than, say, the Space Shuttle's wings are. The "lateral" surface is pretty much inconsequential to this sort of atmospheric drag. Aerodynamic drag is, almost entirely, due to "projected frontal area," after all (By the way, that's the core of why I would object to launching "with pods").
As for heat build-up, that would be no more subject to that problem than any other element of the ship. No more than, say, the vertical stabilizer of the space shuttle.
Yes, it will be subject to SOME additional drag, but it's minimal... it's the mass that's really the concern here. But that would be easy to compensate for as well, simply by having the "underside boosters" be a few feet longer than the "top side" boosters, or by any of a number of other methods.
(FYI, I assume that the "conning tower" is really an antenna array, anyway, not "habitable space" per-se... and as such, it's probably quite low-density compared to the rest of the ship.)
Assembling things in space would help add flexibility to any design.
I'm not sure "flexibility" is really the point here... it's more a matter of "technological limitations" as far as I'm concerned. Maybe that's what you really mean, though... I suspect that it is, though I can't be certain.
It would always be PREFERABLE to have a single, robust, one-piece ship... except, of course, that in "real world" terms, we have to spend a lot of energy in order to move a single gram of mass into orbit. That's the sole reason that we currently do "modular design" of the sort we're familiar with in contemporary spacecraft design Give us a cheap, effective orbital-lift capability, and the designs would radically change as a result.
Which is why I don't think the OP design should be dismissed out of hand, even if I prefer other updated designs I've seen more.
Well, we all post our work here in order to get comments from other folks... and we have to accept the favorable and less favorable critiques with equanimity.
The OP's ship design looks cool, and would work nicely in most "sci-fi ship" venues. My own comments here have been that it neither fits the "Federation ship" style or the "contemporary spacecraft" style... the two forms which we have reason to believe "fit" in EARTH-BASED "treknology."
If you were totally re-envisioning Trek, and having Khan and his group leaving a far-future earth, and being found even further into the far future, this might work. This ship looks like it would fit very well in a "flying alongside the Millenium Falcon" situation. It simply doesn't fit what I expect to see from a "near future spacecraft" concept.