Ok...works!!!
Still got a bit of high frequency noise on the outputs, not a lot, surely not NEARLY as much as before.
The trace coming off the switch @ pin 5 is sort of recognizable, but doesn't resemble the trace in the datasheet. I can see the switch points, but I can also see a sold 3 waves of rings after the switch turns on.
Nonetheless, it works. I get 13.02 volts at both outputs, well within the 5% resistor tolerance, easily trimmed out, but no need since I'll be post-regulating this down to +12/-12 after I raise the output voltage a bit more to cover the headroom for the regulator.
Output load on both rails is 82ohms, ~160mA per rail.
13.02 / 82 * 2 = 317mA load + 2.5mA resistive load for feedback + about 100mA for the LM2588 itself = ~419 mA
13.02 * .419 = 4.45 watts
Input voltage is a steady 4.63v @ 1.81amps = 8.38 watts input
8.38 / 4.45 = 53%
Not great, but not terrible and doesn't bug me since all I want is the voltage and I'm driving this with a 4 cell pk of D's rated at 10000mAh (whether I can actually get the rated capacity, I'd doubt it, but I'll take what I can get).
Top of the head (e.g. bottom of the barrel) math says, assuming full load, max out the output rails at 300mA each (double what it is now), 50% efficient, double the 1.81 amp input, call it 3.7amps, add in another 500mA from the PICs, LCD, other circuitry, 4.2 amps, drawing off a 10AH battery pack, gives me about 2.4 hours with the numbers, maybe 1/2 that, a little over an hour semi-realistically (.4 C draw from the batt pack, inefficiencies, other loads, etc.), on a charge.
I'm good with that...especially since I don't anticipate the end unit to run at full load maybe 10% of the time at the very most...who knows...
Full load would entail the least amount of series resistance on the part I'm testing, all of the opamps putting out maximum current, LCD at full brightness with all pixels lit up, the PICs cranking away.
Now then...
MAN WAS I WRONG! Or rather massively ignorant! I now see, after reading thru the aforementioned documents, what a decent ground plane can do for you.
Even though my circuit is far from optimal (need compensation values tweaked, etc), it works, and I think I can assume it will work well once placed on a proper PCB with a generous amount of copper dedicated for a ground plane only.
Next up...wait for the batteries to fully charge up, add in more resistors to get it to a full 300mA per rail for a load, and see what it does.
(And small booster, low current, <100mA, for a USB charger to kinda trickle the battery pack up a bit while I'm connected to the PC.
Ya, 100mA max going into a dead 10,000 mAh batt pack. That'll take awhile.

Like ~7 days if the batt pack was completely dead, and the circuit was off, and I left it on the USB connector continuously. A lot of IF's there.)
On another side note...on that last try with the circuit, the regulator heated up faster than the resistors and I generally shut everything off once it hit about 160F (100F above ambient). Took about 3 minutes under light load to get to that point.
This time around, with the 82 ohm loads, it barely got warm. Last reading I took after about 15 minutes of running was 80F...20F above ambient, and both resistors were up to about 150F.
It also draws 250uA (assuming that's anywhere near correct with my crappy meter) with the battery connected and the LM2588 "switched" off.