I've chased this on a on a M104 inline 6, I'd expect the operation to be same as a V8 but the hose routing would be different. If all else fails, a work around could be to insert a resistor in line with the O2 signal , a relay that jumpers out the resistor. This relay would be tied to the air pump wire so when the air pump runs, the O2 signal drops.

410 is secondary air injection problem. First how the system is supposed to work:

With a cold / luke warm engine start, the electric air pump operates, switch over valve opens ( vacuum control valve at front of engine ) to allow vacuum to be present at the check valve ( right side by the exhaust manifolds ).

This system introduces air into the exhaust ports to allow burning of fuel past the exhaust valve. It also provides a bit of extra oxygen for the cat converters.

After a few minutes, the air pump stops and vac valve closes. Once the engine is warm, you have made your first stop and the engine is at idle for 5 or so seconds, the air pump system turns on and the ECM looks for the front O2 sensor to drop in voltage due to excess O2. This test lasts about 4 sec and is done every trip as part of the OBD2 cycle. If the O2 voltage does not drop, a code will be set.

Now for the likely fix. There is a white nylon tube that runs under the ignition coil cover. It can become pinched when the cover is installed or it just gets brittle and cracks. A pinch / break here won't allow the check valve to open and allow air to flow.