Of course, the first thought was this is a "Reversed" Transformer (Primary and Secondary input / output points reversed).
This is due to the reference of 3√ė 4 Wire Wye, not to the 120/208 VAC.
With this aside (mention later), the 120 VAC 3√ė 3 Wire Delta is a "Non-Normal" setup - but not something unheard of.
Bjarney's mentioning of NEC's reference to 120 VAC three phase motors
is why this system is not weird (to me).
The Transformer is obviously (to me, again) built to supply power for the above mentioned scenario, along with a few others which may or may not be "obvious" - such as:
* System with maximum of 120 VAC on any L-L circuit, or L-G fault,
* To deal with Harmonic problems - similar to what is achieved with using a 1√ė 3 wire system off a 4 wire wye (via transformer),
* To eliminate the ground loop noise involved with the use of an active grounded conductor in normal circuitry (like the 60/120 VAC 1√ė 2 Wire systems do).
In this case, the system may be grounded or ungrounded. Grounding the system will make the Voltage to ground more stable, so this would be my choice overall (if used for non-critical loads!!!).
Grounding may be done in one of tow ways:
[*] Ground the "Center Tap" of one coil, and run this conductor out to be bonded to all the metallic equipment ony - not an active conductor; this would be "my best choice option",
[*] Ground one active "Phase" line, and use as an active conductor; this would be "my last choice option".
The Wye / Delta setup is really not very odd to me, but as said before, the "4 Wire" reference is kind of strange for a Primary of such low voltage!
Maybe the reference is "Either 120 VAC or 208 VAC" for the Primary input, or just that the Primary is planned to be used on a 4 wire wye system. Either way it's simply a guess to me!
As to the voltage drop problem - it sounds like the Transformer (likely a single coil of the Secondary) is being overloaded temporarily; or the circuit affected is very loaded and the heaviest loads are far away from the source.
Check the KVA of the Transformer, and the loads of the branch circuits.
If the KVA is, for example: 30 KVA, then figure only 10 KVA will be available per coil.
If most loads are across only "One L-L Circuit", such as "A-B" only, this will load up one single coil and result in the problem you are experiencing.
Keep us informed to the outcome!!!