Thanks for all the replies!!
Beats- I used compressed air to blow out the fuel line (air inserted at the carb side of the line, watching bubbles come up from the fuel tank) both for the "ON" and "Reserve" petcock positions. A good technique to have in my pocket, thanks! However, no change.
The carb's fuel filter is clear.
Bluecati- I messed with the air-mixture screw every which way- starting point being 1.5 turns out, I messed with it until I went all the way in and then until 2.5 turns out. These definitely affected the way the bike idled, however, little to no change on the bogging while accelerating under load.
I've had a little over a gallon of gas in the whole time. I got some more gas and filled the tank completely up thinking this was a fuel pressure thing. No change.
I should note this major bogging happens when under load. I can mostly rev it up when the bike is just sitting (though not as well as other bikes I've seen).
I tried to simulate what happens while filming the carb and trying to get it bog out. I couldn't really do it, though I approached a "bog" and the carb spit back fuel. I feel like this is happening while under load- fuel gets spit back and not put INTO the cylinder. But maybe I'm witnessing an Effect and not a Cause.
In the video the air filter and air box cover is off, and I realize that it will not run right like that, but I've been testing it with the air filter/box cover on. It doesn't seem to make a difference at the moment.
If I am very strategic, I can get a good rev going, throw the choke on and accelerate powerfully and even rev up to a decent rpm until I hear it start to 4stroking, at which point I can strategically open the choke up a bit to get maybe a few more rpms, a very clumsy and dangerous dance I do
Oliver- I looked at that link, and it appears that I have the version with the idle jet with a hole in the top, and that particular breathing hole on the carb appears to be properly plugged on my bike.
I am reluctant to mess with jetting because these are apparently "stock" (maybe they're not) and when I woke the beast up, the first test drive around the block ran a lot better than it does now... What the heck did I do

The timing adjustment I made was minor, only about 1 degree advancement.
Scooterhelp lists the following specs for P125x:
venturi size: 20 mm
main jet: 98/100
slow running jet: 160/100
throttle valve: 6823.01
mixer tube: BE 3
atomiser: 280/100
starter jet: 60/100
I assume those numbers are written "out of 100" and that my 'combo' idle jet is "AND". So, 48 AND 160 is my idle jet, meaning 48 starter jet and 160 slow running jet. Not sure if that's the correct convention- I'm just spitballing! I have no idea what my atomizer is, but my main jet is 98, and its holder (mixer tube) is BE 3.
Thanks again for the help. Any other ideas?