The related observation in astronomy is stellar parallax:
the observer knows the Earth's position relative to the Sun, measures tiny annual oscillations in the direction to another star, and estimates the distance to the star.
A star at a distance of 1 parsec (pc) appears to shift by 1 arcsecond (4.85 μrad) per astronomical unit (au) of lateral displacement.
Since parallax is inversely related to distance, the relative error of a parallax-based distance estimate increases with distance.
For any given star, the position uncertainty ellipsoid is shaped like a needle pointing toward the Sun.
Aboard an interstellar spacecraft with limited observational facilities, it may not be practical to measure angles between nearby stars directly.
Instead you might individually measure their positions relative to several distant reference stars near them in the sky.
For best results, you would need to account for the stars' proper motion over the duration of your journey.
After doing that data reduction, you can compute angles APB, BPC, APC fairly precisely.
However, the Sun distances SA, SB, SC have significant uncertainties which propagate to other quantities computed from them, e.g. distances AB, BC, AC.
You can estimate distances PA, PB, PC by the law of sines, but these are uncertain by similar amounts.
Alternatively you can estimate PA, PB, PC by comparing apparent magnitudes to absolute magnitudes, which are subject to the same uncertainties.
Then you can find the intersection of spheres around A, B, C to estimate position P, but still no more precisely than distances SA, SB, SC.
P is not so much a point as a probability distribution cloud which expands as you move away from the Sun.
In the real world, our most distant spacecraft Voyager 1 is only 145 au (0.0007 pc) away.
Within about 400 times that distance, or 1/5 the distance to α Cen, the Sun is the brightest star in the sky by far and would remain your primary point of reference.