Ritual has done some truly terrible math: We know 7 people picked coffee fruit and yielded 175 lbs which when roasted became 35 lbs (15.9kg). We know that a typical worker would've been paid $5.25 for 8 hours of work, but it's not clear if the baristas picked fruit for 8 hours or if they did this for a couple hours.
Now, giving them the benefit of the doubt that they put in a full 8 hours, the labor cost to pick the 175 lbs of coffee fruit would be $13/hr * 8 hrs * 7 people = $728. And that yielded 15,855g of roasted coffee - which would be $0.046 (4.6 cents) per gram in labor cost due to coffee fruit picking.
At a typical dosage of 18g per cup of coffee, that would be $0.83 of that $12 cup of coffee which is due to coffee fruit picking labor costs.
Now, at a typical wage of $5.25 per day, the cost would've been $5.25 * 7 = $36.75. Thing is those are professionals, they probably would've picked more in 8 hours that 175 lbs, but even so if we assume that they are just as productive and would've also yielded 175 lbs then the ratio of wages is simply 36.75 / 728 = 5.05%. Which means in a typical $4 cup of coffee, the cost of the labor to pick the coffee fruit is a mere 0.0505*$0.83 = $0.042 --> 4.2 cents.
Hence... when you want to pay the people who pick the coffee $13/hr instead of $5.25/day, all you need to do is raise the price of a cup of coffee by $0.83 - $0.04 = $0.79. Your $4 cup of coffee when adjusted to San Francisco wages for the coffee fruit pickers would be $4.79. Not $12.
Note in particular that if the baristas worked for, say, 1 hour then the labor cost should've been $91 not $728 which means that the math changes in that the barista-wage roasted coffee is $0.103 per 18g, and the actual coffee fruit pickers would've picked 175*8 = 1400 lbs of fruit, which should yield 280 lbs of roasted coffee. The labor cost is unchanged, though, at $36.75. In this case you also need to factor in production, since the baristas would be paid a wage equivalent to $0.103 per 18g roasted coffee and the actual pickers would be picking at $0.0052 per 18g roasted ($36.75 / 280 lbs). In this case the $4 cup of coffee includes in it about half a cent of labor to pick the coffee fruit, and to adjust to San Francisco wages you simply add the $0.103 - $0.005, making the $4 cup cost $4.10.
Do note that I understand the concept of charity and I understand why one may be motivated to buy a $12 cup of coffee, but let's break this down in another way:
If the labor cost to pick coffee were to jump from $5.25/day to $13/hr and the average worker can pick 4.375 lbs of coffee, once roasted, per day then the labor cost jumps from $1.20/lb to $23.77/lb, an increase of $22.57/lb. This would affect the cost of green coffee in a slightly different way, as green coffee weighs about 33% more this would be 5.83 lbs of green coffee per day. This would mean that a $22/lb coffee becomes a $44.57/lb coffee as the roasting is already done in the US at US wages and the additional cost for loan repayment, rental, equipment, electricity, employees at the roasting facility is already factored into that $22/lb price - using $1.20/lb in coffee fruit picking labor. There are no further adjustments needed here - shipping cost is still the same as the weight shipped hasn't changed.
Here again I don't see how Ritual ended up at $100/lb - I can easily understand how they arrive at $45/lb, but even giving them the best case scenario on their ambiguous document I don't understand $100/lb. Further, I don't see how $100/lb implies $12/cup when $22/lb implies $4/cup.