Ummm - would not have encouraged reading amps to ground at fuses, that is like attaching a ground wire, and could blow fuse upstream or fuse in meter. Amps across fuse holder (one side to other) - OK. Let us look up stream from fuses at the fuse links inr the fuse box under hood. They are like main breakers. One may have popped with engine wire change, some of them run several things downstream.

You should disconnect battery for this test. The put meter on "ohms", short meter probes together, meter should show change from open loop to short (very low ohms). Remove each fuse link (square things with windows in the top) and meter them for continuity. Do one at a time so they go back in correct socket. If they are the bolted in style, you can meter across for ohms, no need to unbolt. Go ahead and check all the push in fuses again just to be complete. See pix.

May have had some built up charge on capacitors inside the BCU that let light work at first. Or new engine harness might be wired just a little off from other. Attached is every wire that is supposed to go to the BCU, as you can see not a lot of things on the engine, but might get from point A to point B by looping thru the harness.

 

Just noticed someting, from you image i see a C0601....I dont have that. What i do have is a spliced wire (blue splice connector) on C0571 (most left brown wire) that is yellow and goes to the starter solenoid? (round metal unit right behind the fuse box under the hood)
I will take a picture of this...see attached.

As for the fuse links, they are all good, incl bolted ones.

 

The C0601 is present on vehicles with Secondary Air Injection, it runs the SAI air pump. As to your modified wiring, don't know that one. You'll need to see where it goes and try to figure what previous guy was trying to do. Next would be check the BCU connector at the module for power.