And for the oil to really work as a transformer cooling medium, the winding would need to be specifically designed for (with spacers forming channels within the winding for the oil to flow through,...).
With this use, you have to make sure no winding is loaded above the current it was designed for. It does not matter what exact function the contraption will have nor how it is connected, nor what real power it is transferring, for the heat what matters are the losses and their vast majority comes from the current passing through the ohmic resistance of the wire the winding is made of. And don't forget there is a square law between the current and dissipated power (Ploss = I^2 * R), so even a seemingly small current increase (e.g. 10%) leads to way larger power losses increase (21% for that example).
Oil cooling of a transformer which has not been designed for it won't help that much. These ballasts use to be designed to be thermally tightly coupled together, so if you manage to add extra cooling to the core assembly (thermall conductive pad between the core and a bigger metal acting like a heatsink,...), it will cool down the winding as well.
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