Is it plausible to use other stars for the proposed FOCAL mission instead of the Sun?

For some time, the far-reaching and speculative idea of using the Sun as a gravitational lens has been floating around. See this and this. This would require sending a spacecraft about ~550 AU of a distance away from Earth, so the idea is not currently realistic.

But I am wondering, is there any plausibility to using other stars (like the Alpha Centauri stars)?

Is there there enough parallax for this to be useful? How far away must a satellite go to be able to manipulate parallax to its advantage? Are there any shortcomings to this idea that make the FOCAL proposition superior? And of course, are there any shortcomings to both propositions?

I can imagine there's not much freedom in where the so-called lens can be used, but maybe the limited field of view can still be useful if the satellite could roam the solar system?

I'm curious if anyone can give useful feedback or calculations.

Gravitational lensing works from anywhere beyond the focus, so in that sense, we could use any star as a gravitational lens. The problem is that the field of view is tiny. We only get any useful information from alpha centauri as a gravitational lens if the target object is almost exactly behind alpha centauri from our point of view. To look in a slightly different direction, we need to move a huge distance sideways.

In fact we do use gravitational lensing from other bodies for astronomy. We use very large clusters of galaxies to amplify the images of other galaxies which lie behind them, and we watch for brief flashes as planets and stars line up for a moment as a way to detect planets (called "microlensing").

Being closer to the focal point is better because it allows more relative movement. You don't have to be right on top of the focal point, but closer is better. Alpha Centauri is very far away.

The focal point is decreased proportional to the mass/radius squared, so, the Sun, about 1000 times the mass of Jupiter and about 10 times the radius, it's focal point is 1000/100 or about 10 times closer than Jupiters. For good imaging with some ability to pick targets and move around, a star or perhaps heavy Jupiter would be ideal, or a white dwarf or neutron star or black hole, but none of those are close. There's a white dwarf in the Sirius system but that's twice as far as Alpha Centauri.

Here's a diagram of the focal points of objects in our solar-system. Density is important too, but more mass generally provides greater lensing, or more light to be collected.

Source of image

If there was a heavy Jupiter, say, 10 times the mass of Jupiter, about 600 AU from the sun, that could perhaps be used, but there's almost certainly not an object that large that close to the Sun because it would have been detected, either directly or by gravitational lensing observation. An object that far in orbit would also provide a narrow viewing range as an object that distant would move slowly through the sky. That's why even the most distant known solar system objects don't work. They're too close. Even planet 9 is too close. A very distant planet or large dwarf planet in the Kuiper belt might work somewhat, if one is found, for example, and Earth like planet about 1/4 light year's distance could work a little bit, but it would provide a very narrow angle of the sky to look at.

Our best bet, given the ability to chose what target we wanted to image would be to use the sun, even if 550 AU is much further than any craft has ever been sent, with the possible exception of the flying manhole cover.

The focal point of the Gravitational lens of Alpha Centauri is a few hundred AU from the star.

But Alpha Centauri is 270000 AU from the solar system. The Sun's gravitational focus is much nearer.

To add to Steve Linton's correct answer, there is actually a specific example of observing one of the Alpha Centauri system stars as a gravitational lens. Lensing of background stars by Proxima Centauri was predicted to occur in October 2014 and February 2016 by Sahu et al. (2014). Zurlo et al. (2018) observed the astrometric shift caused by the 2016 event and used this to measure the gravitational mass of Proxima Centauri to an accuracy of ~40%.