If the only problem is drilling the one hole, just remove the header with a sawzall, drill, replace the header. Literally a five minute job. And no borrowing tools (assuming you have the saw).

The cutting part is easy. You are going to use a blade designed for metal in the sawzall (or one designed for wood with nails). You want to cut the nails where they transition from the stud to the header, without removing much (if any) wood. So, position the saw with the blade against the stud and cut upward between the stud and the header. This will remove minimal wood from the header and allow you to remove the header without damaging it. You will cut the nails on both ends of the header in this manner.

Drill through the stud from the doorway side (where you now have plenty of room).

Reposition the header back where it was and toe nail it on the cramped side. You can either toe nail it on the not-cramped side, or nail it in place using nails through the stud from the wall-side of the stud.

Feed the cable through your newly bored holes and you are done. Time for a well-deserved beer.

Your local code will define nail size and pattern that would be appropriate for securing the header (but since it is not load bearing, I bet it doesn't matter). If the walls were professionally framed, I would just use the existing framing (including the prior nails holding the header) as examples of appropriate nails and pattern to use.