How do I calculate the correct rake for a steep access ladder?

What you're describing is basically a space saver stair or alternating tread unit, and the good news is the geometry works differently than a normal stair, which is what trips people up. First thing to get straight: the pitch doesn't come from how narrow each tread is, it comes from total rise over total run. Narrow treads just let you shrink the run for a given rise, which is what makes the whole thing steep. With alternating treads, each foot gets roughly a full tread depth anyway, because your left foot lands where your right foot was on the previous step. So you can make the treads about half the width of a normal stair and still feel like you're climbing something reasonable.

The old rule of thumb for comfort is that twice the rise plus the going should come out around two feet per step. If your rise per step gets much past eight inches and the going shrinks accordingly, it starts feeling like a step ladder instead of a stair. There's a point where you just have to accept you're building a ship stair and design the handrails accordingly, because people will absolutely need something to hold onto with these. On the stringers, the biggest mistake I see is sawing the tread housings too deep into an already narrow stringer. Go thick on the timber, at least a proper 2x rough size or wider, and cut shallow dadoes rather than screwing cleats on, since the screws work loose over years of foot traffic. Support the bottom end on a plate or floor, don't rely on the angled cut alone. One practical note: in a lot of places anything steeper than a conventional stair gets treated as a fixed ladder by inspectors, not a stair, which changes the rules that apply. Worth a quick call before you cut your nice pink poui, because that timber won't be cheap to replace.

I'd push back on the dado advice. On a steep unit the stringer is working at a brutal angle alredy, and chopping housings into it weakens exactly where the load concentrates. I built a paddle stair for a loft about ten years back with cleats screwed and glued on, and it hasn't budged a millimetre. Glue is the part people skip. Construction adhesive on the cleats plus screws is stronger than the timber around a cut-out, in my experience anyway.

I built a steep alternating set into a basement about six years ago and the thing that saved me was mocking it up in cheap plywood first. Screwed two treads at the planned angle to a wall stud and made myself climb it for a week. Moved the layout twice before I ever touched the good stock. Also cut a story pole for the tread spacing so every notch lined up, which beat measuring each one off the stringer end by a mile. My only real regret is the treads, I went thin to save headroom and they flex underfoot. If your timber allows it, go chunky on the paddles.

This is exactly the problem I ran into with my loft access. The going up part turned out fine, but going down was way scarier than I expected, especially carrying anything with both hands. Ended up putting a proper rail on both sides and that made all the difference. Also worth checking how much headroom you get at the top, mine was tighter than I thought once the angle got steep. Does the pink wood you found hold up okay when you drill into it, or does it split easy? I have never worked with that type before and my first attempt at notching cracked a board pretty bad.

Something else to plan for with that timber: let it sit in the house a few weeks before you cut anything. I made open treads out of some not-quite-dry stock once and every single tread shrank enough to open the joints, close to an eighth of an inch per piece. Stairs get the worst of it too since feet pound them daily. Kiln dried is worth the extra cost here. Also on tread thickness, with narrow alternating treads there's less bearing area at each end so I'd go beefier than normal. Mine were an inch and three quarters and they feel rock solid underfoot.