Easy peasy, just hook it up to hot and hot. Doesn't matter which.
You are relying on the device having competent neutral-ground insulation, but it should, because if it did not, it would fail in any of these applications:
- many European countries use Schuko or other reversible plugs.
- GFCI cares about neutral-ground insulation
- Philippine power works exactly like what you are doing
By the way if you plug it in use NEMA 6 plugs and sockets, never NEMA 5.
If one of the wires is white or light blue, that goes to a hot. Tape it with black tape to mark it as a hot in this installation, so the next maintainer is not confused.
This type of device auto-ranges between 100V and 277V using switching power supply technology, same as most cell phone chargers and laptop power supplies. 100V is Japanese appliance power. 277V is American power in large institutions; it is one leg of 480V 3-phase "wye". It will also play nice with American 120V, Brazil 127V and 220V, NYC 208V, Africa 220, EU 230, UK 240 etc.