Is it possible to indefinitely keep up with the Avatar timer?

The Avatar Project comes with two flavours of progress: facility progress and core progress. It all melds into the same bar, but the distinction is significant.

Facility progress is accrued individually by each facility as they are built and then periodically over time. Taking down one facility removes the progress accrued by this facility, and this facility alone, no more, no less. There is, as far as I know, no other way to remove that progress. The bright side is you can stay on top of it quite easily in the end game.

Core progress is accrued over time or with special events. Completing some objectives will remove some of that progress. Eventually you run out of objectives and the counter will tick inexorably till the end. There is nothing you can do against that in vanilla.

War of the Chosen offers Covert Actions and Resistance Orders to remove some of that progress. Resistance Orders will remove progress every month (but that requires unlocking said order, which could never happen as far as I know). Covert Actions are random, though you probably can get enough of them to significantly boost your game's length. There is however no guarantee you can keep that up forever.

If you want to keep your game alive indefinitely, you might consider cheating shamelessly using the RemoveFortressDoom command. Note that it will only remove core progress, never facility progress.