Is it permissible to go to the mosque in the middle of the Friday sermon due to back pain that makes prolonged sitting impossible?

Attending both Friday sermons is not a condition for the validity of the prayer. Whoever catches one rak'ah with its two prostrations has caught the Friday prayer. It is better to come early and attend both sermons to benefit, and there is no harm in being late for part of the sermon if it is long, due to Allah's saying: {So fear Allah as much as you are able} (64:16), and the Prophet, peace and blessings be upon him, said: "And what I have commanded you, do of it as much as you are able."