Looks like you were busy with it today. During the work week I don't have a lot of time to log in, spent most of the day in the service truck on job sites, which is pretty typical for me, so I log in during the evening when I get home.
Ok, there's just too many things going on here for me to pinpoint what is causing the problem. However, a couple of things to consider. 2001 would be one of the early G Series machines, but I'm near certain that Bobcat no longer used a seat switch when they went with the G Series, they used the on the C Series and some of the F Series machines, but they were problematic. By the time they produced the G Series they pretty much settled with using only the seat bar switch. Now the seat bar switch in one of those is a Hall Effect sensor/switch, no way to test it with typical test equipment, but Bobcat has a tool/tester that will test one.
If that seat bar sensor was faulty, even intermittently, yes, it would lock out the lift/tilt functions and even the drive, just like it would if you were to raise the seat bar. However, what it would not do is shut the engine down, that's the part that has me puzzled, especially if it's not a fuel supply problem shutting the engine down, which shouldn't be likely as you just replaced the fuel supply line.
Assuming it's not a fuel supply problem shutting the engine down, rather, it's an electrical supply or ground (earth) problem, then it has to be a problem that's not part of the control system (sensors/switches) of the machine because obviously the ECU isn't seeing it, hence no beeps. To recap, if it's an electrical problem shutting down the engine, yet you get no beeps, that's typically an electrical supply or ground problem.
Things to to do:
1) For sure check the battery connections. Unless they look squeaky clean, don't just inspect them, actually take the battery connections loose, clean them, and re-connect them. And be sure to disconnect those ground wires that connect to that stud on the negative battery terminal, make sure those grounds are clean and connected good.
2) Until we figure out this electrical problem, if in fact that's what it is, that machine should have an external fuel shut down solenoid. It would be mounted on the injection pump and have a linkage that operates a lever on the side of the injection pump. Start the engine, that makes the fuel shutdown solenoid pull the full lever on the injection pump into the run position. Take some tie wire and tie that lever to where it will remain in the run position, that way when the problem surfaces it won't kill the engine.
3) If we do have an intermittent seat bar sensor problem, which I think maybe not, but I'm not sure yet, nevertheless, let's try something, just for the hell of it, the next time it kicks off the 4 BICS lights and locks down the drives. We've got the fuel solenoid tied back to keep the engine running, now raise the seat bar. With the seat bar up and engine running, push the Traction Lock Override button and see if it will unlock the parking brake and allow you to drive the machine. The lift/tilt won't work, but you should be able to drive.
4) Lastly, raise the cab and check the C407 harness connector that connects the cab harness to the main frame lower harness. With the cab raised, standing at the front of the machine and viewing toward the machine innards (to the rear), you'll notice a bundle of harness coming from the back rear of the cab and going down the back bulkhead of the main frame. Follow that bundle of wiring down until you come across a harness connector, that's the C407 connector. Take it apart (disconnect it) and inspect the pins/sockets inside the connector, look for burnt (arcing), damaged, or corroded connections. Look especially at wires that are red or red/white, numbers would be either wire number 1200 and/or wire #1900, those are main power wires going to and from the cab BICS panel. You may want to also pull the BICS panel in the cab and look for any problem wiring or connection problems to the back side of the BICS panel.
Let me know what you find.