This may help to understand the grounds.
In aftermarket equipment, there is 1 ground. Ground is ground is ground.
In the Toyota system, there are 3 grounds. Not all the same.
1. Chassis ground
2. Head unit to amp ground
3. Speaker low level (signal) ground
This is the problem you're facing. The aftermarket amp can only work off one ground. No way around that I'm aware of.
The OEM head unit must use it's own speaker low level ground (which is NOT chassis ground) in your low level signal. No way around that.
You can't use that low-level ground to power your new amp.
Now we're stuck.
I think your best course might be to re-install the factory amp and use the high level outputs from that factory amp as inputs to your dsp amp. Hopefully the DSP amp is ok with that high level (-) signal coming out of the factory amp. We know it is NOT ok with the low level signal. This option would leave you with unused RCAs on that Metra harness and that's ok.
The shop would need to go back and re-connect all their bypass work, which is a bummer. When I did my bypass, I specifically avoided any cutting of wires and used a connector with the same terminal type to make it all reversible if needed.
**Edit: Oh, and that black with white wire appears to be the amp shield ground. It should be connected to the head unit SGRN or chassis ground. I'd try both and see if either magically solves the problem. I suspect it won't, but it's definitely worth a shot.
https://pdf.crutchfieldonline.com/ImageBank/v20170313094500/Manuals/120/120708113.PDF
I do wish they would've left the factory amp in place and bypassed for simplicity sake, especially now that there are several wires not being used that mus serve some kind purpose. My MFD touch-screen beep is barely audible now, and I also can't figure out if I can address that somehow. Do I need to tie the GREEN "BEEP" wire in somewhere. I wonder if it's worth reinstalling the factory amp and bypassing?
Surely others have had good success with factory HU + New Speakers + New Amp though?
Some people have said it worked, but I suspect those people all have some sort of ground loop isolator or are just OK with (or don't notice) the ground noise buzz and hum. For a high quality setup, I don't see how you could use the OEM Toyota head unit with a normal aftermarket amp. Same goes for the OEM amp with aftermarket head unit. The Toyota design its own little eco-system of grounds and I've never seen an aftermarket amp setup to handle that sort of integration.