Insulation basement (Rigid Foam). What to do in the unfinished area
When I partially finished my basement, I insulated the exterior walls of the finished part, and left the interior walls and the unfinished without any insulation. My AHJ (in Ohio) approved this, stating that there was no need by code to insulate the unfinished part, nor insulate between the finished and unfinished parts. He did insist on insulating the rim joist in the finished part, which is very sound advice - that is the bulk of heat loss in a basement (unless your frost depth covers most of the basement height).

If your AHJ allows for insulating only the finished area, I would do just that, and leave the walls to the unfinished area without insulation.

In my basement I have around 420 sqft finished area, an adjacent 320 sqft unfinished area and another walled off unfinished area of 220 sqft. The finished area has a single heating vent at the ceiling. The finished area is slightly cooler than the rest of the house, but is noticeably warmer than the adjacent unfinished area. The walled off unfinished area is again colder than the unfinished area that is next to the finished area, so the heat gradient is fairly obvious. However, even during the coldest weather the coldest part of the basement is still fairly warm, and with a frost depth of only 32 inches my experience might not apply to colder climates.