67 Barracuda Dash harness install

Much easier to remove the instrument cluster. Just take your time.
What Joe wrote is good advice. Removing the seat will provide more room. Lowering or removing the steering column will make it easier to work on the instrument cluster. In fact, IIRC lowering it is a must to provide the clearance.

1967 Barracuda had a supplement to the shop manual with its own electrical diagram. The difference between it and the service manual Valiant diagram are minor but obviously they put in the sup because there are some. 1968 Barracuda uses different connector cavities and so is not interchangeable with '67 without modification.
For some ref photos, here's a picture of the bulkhead side of my '67 notch back with two of the wires removed for inspection.
Wires at the bulkhead are all original; alnt 6, 2 speed wipers, no A/C or light packages. The white wire and bundle in the lower right is not - those are for the aftermarket tachometer.
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Engine compartment side: Repop harness pruchased from YearOne over 20 years ago. (Gray cable is for MSD ignition.)
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Don't get put off track by MAD. They over generalize. A link to another perspective.
The condition of the wires in your picture show the problems on your car are not only in the bulkhead connector, but also the condition of the copper wiring. The wires have lost insulation and the copper is green oxidized quite some distance in from the connector. Electrons travel on the surfaces - so oxidation like that diminishes their capacity.

The electrical wiring problems in my '67 (which was an AZ car before coming to the northeast) were at the headlight switch connector, the bulkhead, and the headlight connectors. (If you plan to drive at night more than just once in a blue moon, a headlight relay system is will take most of that electric load off of the bulkhead, headlight switch, and wires. Crackedback makes one that is plug and play. )

Is it worth buying a repro harness for under the dash? In my opinion, if you can find a good quality one made specifcially for a '67, yes. If you can't, then lay yours out on a big board, invest in the right crimpers and terminals, and fix what is needed. That's a big initiation but is doable if you have to go that route.