The problem I see personally* with making this a pre-Constitution is the same problem I have with trying to work most of Enterprise's "plot" points into any view of the Star Trek Universe that includes TOS--Enterprise just has too much damn technology.
The technology used in Enteprise is fundamentally the same as that of TNG, DS9, and/or Voyager. Some of the terminology is different ("phase pistol" instead of "phaser," "polarized armor" instead of "deflector shield," etc) but it all seems to work 100% the same way. They have technology that they're not supposed to have even according to TNG--the Daedalus-class Exeter, lost in 2167 or thereabouts, didn't have subspace--and of course TOS didn't have "real-time" subspace comms with Earth the way that NX-01 does. Based on those examples--and plenty more that I won't go into--I simply despair of ever integrating Enteprise into the timeline of the first four shows, no matter how many Bobby Ewings jump out of the sonic shower to press the magic rewind button.
That's why my suggestion would be to just shunt this ship into a parallel reality, the one in which the NX class seems to have replaced the Daedalus as the gold standard of early starships, and then come up with a whole fleet's worth of new designs to populate that reality. One can only imagine, for example, what an Enterprise-ed Loknar might look like... or several of the odder FASA designs that fall within the "flat" design style that Enterprise is giving us...
Who knows--a bit of Enterprise-ing might even make the Akyazi tolerable!
*One of the problems with this sort of discussion, of course, is the fact that we all create (or re-create) Star Trek in our own image. Even before Enterprise took a quantum torpedo-sized hammer to the timeline, there are enough inconsistencies and continuity glitches in the first four series that, as a fan, you have to come up with your own interpretations and explanations for them. So, when I talk about stuff like this, it's all filtered through my own "world" view, which I fully realize is quite different from (I would guess from past experience, anyway) most other people's.
I'm not trying to say that mine is "right" and/or that anybody else's is "wrong" here, just trying to point out where I'm coming from...
One more thing that I'd say about the ship seen here would be that, on further study, it seems a little too much like bits and bobs of NX-01 taken and rearranged without the sort of significant changes that one would expect from 80 or 90 years of technological development. For example, the nacelles of the Daedalus and the Constitution (particularly the early "Cage" and "WNMHGB" versions) are certainly similar--but not exactly the same, as these seem to be.