So, I successfully flashed the BIOS with some skanky ROM I found sleazing around teh webz (actually, at Anorak, so probably had all its shots, etc.). There was a real heart-stopping moment when the progress window for ATIWinflash said "Not responding" about 3 bars in... I decided to go do something else for a few minutes and when I came back it reported success. Here's the process I followed in case it helps anyone else:
1. Set BIOS switch to position closest to video port side of card (in my case, mining/clock speeds/etc. was exactly the same with either BIOS loaded, so I concluded that the same BIOS had been [inadvertently?] flashed to both positions).
2. Save BIOS using ATIWinFlash 2.7.7
3. Load "powersave" ROM for my specific card (XFX RX-570 Black Edition) onto ATIWinFlash and hit the program button.
4. Skip over the panic attack when progress window says it has stopped responding.
5. Reboot
6. Run ATI Pixel Clock Patcher to sign custom BIOS.
https://www.monitortests.com/forum/Thread-AMD-ATI-Pixel-Clock-Patcher7. Reboot
8. Profit
Now my card is running Ethash at 25.4 MH/s (using Ethminer 0.12.0) with MSI Afterburner reporting a core clock of 1050 MHz (down from 1286), a memclock of 1900 MHz (up from 1750), and a card power draw of 65W, down from ~100W (this, however, is not accurate, as the total current on the PCIe connector [yellow wires only] is 8.5A, down from 10A, and 8.5 x 12 = 102W right there; still, can't argue with the system power draw going from 400W before to 300W now...).
So these are pretty decent results, but not too different from what I was getting with the other RX-570 right out of the box (24.8 MH/s and a system draw of 300W), so I'm thinking that the custom BIOS I just loaded is what was supposed to be installed for the "mining" position of the dual BIOS selector switch... And I suspect that further monkeying around with the straps won't get me all that much, but I'll likely try the "One Click Timing Patch" feature of Polaris BIOS Editor anyway, I just want to make sure this BIOS runs okay for a couple days straight, first.