tl;dr: It could be dozens of years from now or it could be next week!
Since the artificial satellite orbits are constantly changing, what's necessary is to regularly run a conjunction search program that updates itself with the latest TLEs for the satellites, and uses them to look far enough ahead that you can book a flight.
As your map tells us, two different orbital planes (which intersect Earth's center) will always intersect along a line which intersects Earth's surface at two points.
Since we can already guess this will be at most a rare event let's approach this statistically. Instead of tracking the motion of three objects far into the future (which is not accurate for low Earth orbits) let's just assume that the four satellites (three artificial and one natural) are smeared out along their orbit; every 1° of orbit has about 0.28% of the satellite in it on average.
Now low Earth satellite orbit planes slowly but steadily precess around the Earth. This is why the space stations have
These are rough figures:
i (°) a (km) ω_p (°/day) period (m)
ISS 51.6 6378 + 425 4.9 93.0
Tiangong 41.5 6378 + 400 6.0 92.6
Hubble 28.5 6378 + 540 6.9 95.4
The Moon 26.1 [1] 384,748 0.05 [2] 39344.
[1] 23.4 ± 5.1° at 18.6 years, https://en.wikipedia.org/wiki/Lunar_precession
on it's way to maximum in March 2025 https://en.wikipedia.org/wiki/Lunar_standstill
[2] the 18.6 year precession is due to the Sun's gravitational perturbation, not the Earth's J2.
$$T = 2 \pi \sqrt{\frac{a^3}{GM}}$$
$$\omega_p = -\frac{3}{2}\frac{R_E^2}{(a(1-e^2))^2} J_2 \omega \cos i$$
using $GM$ = 3.986E+14 m^3/s^2, $J_2$ = 1.0826E-03 (unitless) and $R_E$ = 6378137 meters.
But before we get too involved in that
let's just try to estimate the chances that two artificial satellites in LEO can appear to be within a 0.5° circle in the sky as seen from Earth's surface. Let's choose an average viewing distance half-way up to the zenith at 600 km.
The first artificial satellite probably spends 1/3 of its time between the Moon and some place on Earth's surface that can see it.
There will be reduction factors for day vs night viewing (transits across the unlit part of the Moon during the day will be invisible) cloud coverage factors and absence of landmass under your feet. So let's just use 1/20.
The second satellite's orbit covers a substantial fraction of a $4 \pi$ spherical shell around Earth, $i=45°$ covers about 70%. That's 4E+08 km^2!
At say 500 km the Moon's circle is only 5 km in diameter, so has an area of 15 km^2. That's our target, about 4E-08 of the sky where the satellite can appear.
A transit conveniently for us about 1 second. Our visibility factor of 1/20 means the rate is 0.05 transits per second somewhere! Multiply that by the probability that the second satellite will happen to be in the same circle, and we've got 2E-09 double-transits per second, or about 0.06 per year!!
It is absolutely worth it to do a more careful calculation!!
This is all I can do for now, but the back of a cow-shaped-envelope calculation suggests that the rate is within reach.
It could be dozens of years from now or it could be next week!
Since the artificial satellite orbits are constantly changing, what's necessary is to regularly run a conjunction search program that updates itself with the latest TLEs for the satellites, and uses them to look far enough ahead that you can book a flight.
The area
That's a ratio of 5E-08. The satellites move through the target in about 1 second, so on average for one satellite the time between passes through our moon target is 18,369,104 seconds or 212 days.
So the ballpark
