There is no Power Source that is only 'live' when the engine is running;
You have constant power sources or ignition powered sources.
take a relay and hook up the ground side of the relay apply to the wire going to the oil sending switch so when the oil switch grounds it will apply relay.
This is creative idea - unfortunately the logic is inverted, the Oil Switch is 'closed' when there is NO pressure (i.e. when engine is NOT running) and open when there IS pressure.
But along those lines, if you had the Coil Negative (85) indeed to the Oil P switch, then ignition-switched Power to the Coil Positive (86) AND to the common 30, then the normally closed contact (87a) to the device, that would probably do it.
The Truth Table would be as follows
Ignition Off - Relay is off, so no power consumption by relay coil with key-off - and no power out
Ignition on (engine not running) - Relay energizes, but 'output' would be to the N/O contact=> no power out
Ignition on (engine running) - Relay turns off, you get output from the N/C contact
You would need to be sure you pick an ignition supply that has adequate power capability (and there probably isn't one) so that means an additional relay.
The downside of this this (above) is that it is going to come on pretty much as soon as the bike is started (indeed it could possibly even come on while it's cranking)
There are other more sophisticated ways to do it, but this should be within capability of most
Alternatives would be electronic comparator circuit or using the stator output to enable a trigger
Here's another way you could do it - I designed this initially as a delay circuit for turning on headlights;
This would equally work for your amp, it only turns on after you operate the brake the first time
(Again you need a switched input so probably need a second relay to supply that in place of the "headlight Power" input)