Certificates of airworthiness..

[B61]

I am a bit out of date here....

There used to be Private and Public Transport CofAs, but now there is just the non-expiring EASA CofA.

If you have an aircraft with an engine that gets to 2,000 hours, can you just change the ARC to a non-public transport ARC, and carry on using the aircraft "on-condition" when it reaches that point, and just not do any training on it ?

It is an old airframe that might have a couple of years more life prior to scrapping....not really worth a new engine in it.

[justmaybe]

There is no longer private or public CoA cats.
In terms of the engine continuing to be used in an aircraft for aerial work that consists of flight training, have a look at CAP 747, and in particular Generic Requirement 24, and that will give you a good heads up