Underpin as usual,pour footing with weld plate in ajoining wall(s), Fasten drain tile in position roll in Precast sections with waterproofing and miradrain attached, wedge in a lap splice at joint, repeat as necessary...
The caulk joints between panels need a little TLC, damp pack the base with low expansion hydralic cement, I' make the panels a few inches short for ease of installation and infill the void with brick.
Quick and "Dirty" lay the wall out of stack bond CMUs grouted with necessary sesmic rodding(threaded 4'long horizontal bars?), waterproof best you can. Unroll/fold as you go tacking the top edge to the sill plate a continous piece of 40-6omil EDPM roofing as a water barrier--backfill with washed rock to near grade land scape cloth, dirt, after wall is loaded/braced.
If you pour 5500 PSI breaking strength concrete its almost impervious to water, just install a Bentonite filled water stop in vertical keyway. High early strengh cement would allow daily pours, you could dig from both ends if so lay out an odd # of panels to shorten the process one more day. Glue the rubber roofing together in the final hole. I've had excellant results with edpm on buried CMU vaults.
A single post tension cable/treaded vertical bar in each panel anchored to a footer with more "heel" would increase the ability of the infill to resist soil pressure.
Are you undermining the neighbors footing by more than the width of seperation? If so better call a pro. Or a lawyer.
Pouring from the floor above if possible really cuts the labor of filling the form by hand, if you're also remodelling the first floor.
On further reflection, @ 10" from the neighbor's basement, engineering EQuake wise you're melded to their foundation, especially if you're "downhill" get a real underpin/ sesmic pro....