Ecobee 4 installation - No C constant 24V option on Furnace
Tap the other transformer terminal

The transformer terminal on the right seems to be connected by a brown wire to R as you describe, so clearly the other transformer terminal on the front of the transformer (the left-hand one, with the purple wires hooked to it) is the C terminal.  This is consistent with the wiring diagram you posted, which has wires from the transformer common to the common side of both relay coils.