If it runs with power jumpered to Fuse 4, then the Fuel Relay coil must be the "buzzer" (the Main Relay coil feeds the AMM and ECU, no effect on pumps, injectors, etc.).

If you're sure your relay(s) are OK, about all I see is the Fuel Relay ground (Bl/GN wire on 87-2), which gets grounded via ECU pin 17, but ONLY if the ECU sees the "heartbeat" that rjyousey mentioned, which means pulses from Ignition CU pin 8.
--
Bruce Young,
940-NA (current)
'80 GLE V8 (Now gone)
'83 Turbo 245
'73 142 (98K)
'71 144 (track modified--and going to be crushed unless...)
New 144 from '67 to '78
Used '62 122 from '63 to '67

I've had all sorts of relay problems. Normally the root cause is the tank pump either quiting or the hose hanging the pump to pickup rots or splits ... with out prepump - main pump draws to much current being overworked, excessive current overheats the relay. Doesnot sound like your case although I've heard pump relay buz... why? mmm. One other related that took a long time to figur- car would quit on occasion and alway restart if key was reset- root cause here was bulkhead connector to the left of wiper motor ... just above rear of engin valve cover. This bulkhead has been or went bad on amost every 240 I've owned - inspect very carefully. Best way is to ohm meter between ballast resistor and coil, should be below an ohm I would think, tap or wiggle the bulkhead connector - any ( even slight ) change in resistance indicates connector is flaky .. I generally cut the offending wire on both sides of bulkhead and use a good crimper and stake on to recconect. This is a critical connection since its supplies several amps of current to the coil - a drop in currnet ( connection opening ) will cause the spark to cease from coil and the heartbeat signal supplied to relay will come up missing then relay drops out and you have to restart the engin for another go. The bad connection can cause the craziest symtoms ... maybe even a buzing relay - what got me was the would more often then not quit in the middle of a corner. My case - one - the connector was already repaired by bad lets say mechanic... he used crapp crimper and the wire was very loose in the connector ... basically a very slight sideways force on the connector would open the connection.