It's true that Hubble can't take images close to the Sun for physical reasons (risk of damaging the telescope). But note that, as I pointed out above, the ESA used Hipparcos to do exactly what you're suggesting: verify that the Sun's gravity changes the apparent positions of the stars that we observe by the amount predicted by general relativity. It's just that Hipparcos measures star positions so precisely that it's not necessary to wait for a solar eclipse and observe the stars closest to the Sun at the time. Instead, you can see the effect in all the stars that Hipparcos observes, with the size of the effect depending on how close to the Sun the line of sight to the star is when observed (in fact, Hipparcos measures positions so precisely that we must take the general relativistic effect into account when using it's measurements, in the same way that we essentially verify predictions from general relativity every time we use GPS).
Conserve energy. Commute with the Hamiltonian.