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

Whether you use the factory muticonnector on the bulkhead or a grommet is up to you. More important is good support without strain. Connectors also are good locations for test probes.
There must be a route into and out of the passenger compartment. Power has to get to the key switch.

For example, with the key in start, current can flow through the ignition start (J3) and the starter relay wire (S1).
Assuming the neutral safety is grounding, current from S1 flows through the relay's electromagnet to ground.
In your case J3 needs to go to whatever ignition and EFI relay is in the Dakota harness.

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Dome light, Parking Lights, hazards all got power from the 'always hot' buss in the fuse box.
Wipers, turn signals, radio get power from the 'accessory' feed off of the key switch. Some are fused, the wipers have circuit breaker - usually in the wiper switch.

Hopefully that makes sense as to why there must be a power wire going into the passenger compartment, and at least a few coming back out. With relays the electrical load will be less for those just being used as triggers.