The Journeyman by Leon de la Garza (2015), also appeared on Reddit two days later, posted by u/Writes_Sci_Fi
Below are quotes from the story that match your description:
old man who transfers his consciousness to a synthetic body
“Oh well… I’ve been hearing that the only option for us old folk are the synthetics. If I had a choice I think I would have gotten the organic, but the synthetic will do.”
moves to Mars
He left Earth and never went back, and though the ride to Mars was comfortable he noticed the stares from the other passengers, the organics.
falls in love with another robot.
Paul walked alongside Loren, the silver synthetic that had saved him from his wandering trance, to the city of Mars One.
The female robot eventually chooses to die
With a quick motion she slid her memory module out of place and Paul watched it drop next to her. The body that previously belonged to Loren became stiff, its eyes lost their glow and it fell to side of the mountain.
The old man also became some sort of scientist and invented FTL travel,
Paul discovered intergalactic travel and he left them all behind.
the story ending at the death of the universe.
For the first time in fifty billion years Paul looked at the sky above him and saw only darkness. On the other side of the world he inhabited, the last star of the universe shined.