1994 BMW 840Ci with an ABS problem

1994 BMW 840Ci mileage: 83,000. I'm the unfortunate owner of a BMW 840Ci, with an ABS problem. On starting the engine the ABS carries out its usual test and both the traction control and ABS lights go out. As the vehicle accelerates away and reaches above 5mph, both lights immediately illuminate and stay on. The fault can sometimes be cleared by switching off the engine, restarting the engine and then counting to 30 before engaging reverse, then reverse the car with full left hand lock on the steering wheel! The car will then drive away normally without either of the ABS lights illuminating. If however either of the rear wheels are spun, i.E. On gravel, then both lights will immediately illuminate and stay on. The car has been into the local BMW dealership for diagnostic checking, where they informed me that the pump relay was faulty. This has been replaced to no avail (and also the memory reset). They are now talking of requiring to test the car further for up to a day, at $85 per hour, just to find the fault! Prior to sending in the car initially, I removed all four wheel sensors and checked each of their resistances, which were all comparable. Although one on the left rear was found slightly bent due to stone damage, it was easily straightened. All four wheel segment rings were also cleaned up and in good condition. Any positive advice of tests that I could carry out would be greatly appreciated, before breaking the bank. Thanks in anticipation.