I feel like I've taken a crazy pill. I measured the voltage and current at four points in the circuit. First, at the wall. Next, between the two transformers. Then, after the two transformers (the input to the bridge). Finally, the DC output across 22.5k.
At the wall: 150mA AC @ 122VAC (18.3W RMS power)
Between the transformers: 460mA AC @ 15.6VAC (7.2W RMS power)
At the bridge: 13.1mA AC @ 115VAC (1.5W RMS power)
At the load: 6.2 mA DC @ 154 VDC (1.0W)
That's an immense power loss. The first transformer is losing 11.1W and the second transformer is losing 5.7 watts! And then the bridge (and maybe the capacitor) is losing us another 0.5 watt. No wonder this thing is not performing the way I expect, especially when I load it down with more current and the voltage output drops. Maybe these transformers are just crap?
I loaded the output down with 1.52k (four resistors in series, each 25W) and measured an output voltage of merely 118VDC. That's terrible and that's just about halfway to what I really need.
The reason that I chose this step-down step-up configuration was for safety, so that you'd never have to plug in something lethal. The lethality would come after the step-up transformer.
On the other hand, we plug in 120VAC appliances to the wall all the time. Perhaps I should start thinking of just using a 1:1 isolation transformer and live with the fact that both sides of the transformer are now lethal.