Can you do that? Use a transformer above its listed applied voltage? Legally??
Most 'low voltage' transformers have _insulation_ rated for 600V. So the question about properly applying the transformer is to make sure that the coils are seeing the correct 'terminal to terminal' voltage, not necessarily the expected 'terminal to ground' voltage.
The common situation where this is encountered is a 'buck-boost' transformer. For example, a transformer nominally rated 120/240V primary and 16/32V secondary is commonly used to boost 208V up to 236V. This means that the '32V' coil is actually being used at about 120V relative to ground; but you only have 28V across that coil.
In this case, three single phase 5kVA 208V:120V transformers could be banked together in a wye autotransformer configuration, to give a 21.5kVA bank. You would put the 120V coils in wye on the 208/120V supply, so these coils would see their correct voltages. The 208V coils would develop 208V by transformer action. These would be placed in series with the 120V coils, giving a wye output with a 338V L-N voltage, or 568V L-L.
The output of this transformer is _not_ a separately derived system, so grounding requirements would be different.
What I don't know is the _legality_ of this installation in terms of OCPD and grounding requirements.
-Jon