"Escape route" as a term is fine and used commonly.
Overspeed (improved performance), equally, can be calculated manually, provided you have an AFM with the relevant data.
By the way, as demanding is to fly an early turn
Which is why the takeoff has to be briefed before you launch so that the crew know what they are intending to do. This doesn't protect you against every eventuality but reduces the likelihood of surprises.
I don't know how the modern computer programs run the calculation but I have a suspicion that they don't move the end of the TODR as the weight reduces
While one could use such a technique I see little reason for it. Whether running a manual or computer analysis, it is easy to incorporate the distance changes. After all, the aim is to extract the maximum weight for the conditions (I note that you, being Brit, use "mass" - apologies for our Antipodean slackness in using "weight").
but keep the obstacle distance fixed from the end of TODA
It is but a doddle to run that as the distance from brakes release.
maybe I will have to chat with our company perf engineer
That would be the preferred approach for a specific aircraft and analysis protocol. Doesn't surprise me, though.