The minimum distance required between handrail posts and opposite wall

When concrete is poured so that it will be in contact with a wood part of the building allowances should be employed so that moisture is prevented from becoming trapped and is able to escape quickly. The wood wall that abuts to the concrete stairs should have been isolated with a vapor barrier. The most commonly and easiest to use would be 30 lb. roofing felt tacked along the stair/wall face. Metal flashing and self-sticking flashing would also prevent water from contacting the wall. An isolation or expansion joint would have helped any water that did find a way in to eventually evaporate. Repairing the concrete from contacting the studs is difficult. If there was any space between the surfaces a barrier could be retro-fitted. I'm not sure how feasible it would be to start cutting back the edge of the steps to create a space for repairing the wall? If you could that would enable the installation of treated lumber and water shields (flashing and felt). If not than all that can be done is to silicone caulk the wall/concrete joint, but water will still get in through the concrete by condensing.

I think I'd go with part of my original answer, but leave the railing as-is. Take the balusters out for all treads above the new wall. Replace the treads and risers with longer ones that extend all the way to the wall, closing the gap. Replace the balusters. Done deal. I wouldn't even mess with the framing.

You'll have a little bit of wasted tread space along the wall, but it'll be much less odd than a deep gap. The lowest riser would notch over the wall over the new doorway. It looks like a very clean situation from here.

[original answer]

Were it my home I'd return the handrail into the wall and widen the five upper stairs to the wall, filling in as appropriate above the bathroom door casing to the first wide tread.

I can't see what's going on with the soffit/ceiling above the lower hall, but you might also be able to extend the wall above the bathroom door to meet the stairs, and either terminate the railing there or run it past/above the wall, which will either have a ledge at the top or run all the way to the high ceiling.