There is no way to "reconcile" the differences, the two types are totally different in form and function in too many fundamental ways. The showrunners recognized that at least, and made a dialog kludge to cover it by saying they simply did not use the same uniform in the division those heavy cruisers belonged to.
It is a solution as awkward and ugly as the DSC uniforms themselves, but it does cover the problem more or less since no other Starfleet ships were in The Cage so there is nothing in the original material that directly contradicts the kludge. Anything "in between" the two uniform types would be a third type further knotting up the original compatibility problem.
On the other hand, in the sense of STO with its different uniform for every ship style those are not bad considering there is no long-knap velvet texture available for anything but the medical TOS uniform (and even that looks like a rather iffy compromise between the velour and the double knit materials).
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It is a solution as awkward and ugly as the DSC uniforms themselves, but it does cover the problem more or less since no other Starfleet ships were in The Cage so there is nothing in the original material that directly contradicts the kludge. Anything "in between" the two uniform types would be a third type further knotting up the original compatibility problem.
On the other hand, in the sense of STO with its different uniform for every ship style those are not bad considering there is no long-knap velvet texture available for anything but the medical TOS uniform (and even that looks like a rather iffy compromise between the velour and the double knit materials).