I'll also calculate the area of the segment explicitly using integral calculus, prolly in Mathematica, as AI is very prone to errors as you said. Then we could compare the results.
The numbers, after calculating the expressions by hand and using online Wolfram Mathematica for integral evaluation, turned out even 'better' than your AI suggested. The area of the orbital segment around the perihelion, from entry point at 187 AU til exit from heliosphere at 156 AU is cca 10888 units squared, which is about 1.099×10^(-5) part of total area of the orbital ellipse.
For total orbital period of 27 My, this gives that the time of companion's passage from entry to exit would be roughly 297 years, which is in excellent agreement with what's written in "Help is on the Way" SOTT article. Using the same area method and extrapolating from the exit point, the companion would now be at 250-300 AU from the Sun, only entering into Oort cloud from the 'inner' side of the solar system, which explains why the C's said that it hasn't passed through it yet on its way out.
According to internet sources, brown dwarves have notable (electro)magnetic fields, and interaction of two 'heliospheres' then goes hand in hand with diminished number of sun spots as indicators of solar (electro)magnetic activity, i.e. 'close' interaction with brown dwarf's (electro)magnetic field can easily 'ground' the Sun in the sense of stabilizing its field. With the strong companion's e-m field, the effective central mass of the binary system can then be such that Kepler's 3rd law is satisfied along the whole orbit, in line with what the 2nd law about the areas states for the time of the close passage during those three solar minima.
Linking global drop of temperatures on Northern Hemisphere, the Little Ice Age, to low numbers of sun spots during those times is a bit trickier, as according to our current scientific understanding of electrodynamics, the e-m radiation coming from the Sun would not be affected by outside solar e-m field. Modern example of increased solar activity during second half of the last century got associated with lower global temperatures during those times when compared to lower solar activity during last few solar cycles, which kinda suggested the trend in the opposite direction. In that sense, it seems we're still missing something to properly understand how all these things are related, starting from basic solar physics and stellar energy production. Or, maybe more appropriate term to use would be what the C's once said, it's not missing, but just not yet discovered.