They would certainly need to get the re-fare to Club Europe (business) from the mythical Back Office, but if you don't like it then you just ger reverted back to your old booking, no costs involved. While this happens the BA App may show either 2 sectorss or a CE sector instead of a single EuroTraveller sector. But there is no downside to asking for a refare. If you don't like the upgrade cost (a) consider upgrading with Avios if you have a stack of them and there is availability; or ask at CAI if they have an upgrade price on the day of travel.

Note Economy Plus isn't materially different from any other seat on the aircraft, it just allows you a baggage allowance and greater ticketing flexibility. Club Europe would give an empty seat, catering, more back, more AA miles, and a slightly better seat compared to some EuroTraveller seats at the back. Most passengers probably wouldn't notice.