Do I really need the wire from the alternator to the bulkhead connector?

The factory system works great. Reduce the load to the bulkhead with the headlight relay conversion and you are golden. The alternator input will never overload the circuit, it just kind of floats along. The fuse block protects the corresponding circuits. Its the battery side that burns stuff up.
I forgot to add in my response, I did just that with @crackedback's headlight wiring kit. Money well spent.