That should be a really good way to check our models are accurate!
But could a photon make such a journey? Yes. But you'd never be able to recognize it against everything else.
[1] https://en.m.wikipedia.org/wiki/List_of_nearest_known_black_...
By measuring the time delay and the separations between the images you can get an estimate of the Hubble constant.
It's really cool to see all the additional, older galaxies visible in JWST's infrared which were not previously visible to Hubble!
[1] https://www.discovermagazine.com/the-sciences/how-the-1919-s...