'64 Dart 270 Wiring

Or should I scrap the idea of buying a built harness and construct one? I have enough electrical knowledge to wire everything I think.
based on what you've written, I'd say you could with some studying and planning. At least a couple areas of understanding will go a long way up the learning curve. Then you could build your own. Slightly easier may be salvaging what you have. For that part, remove everything and tack it to a big plywood board.
Should I just wire everything by hand, attach it all to a modern fuseblock, use the factory recommended fuse amps, and then go with whatever would be recommended for the current stereo (as opposed to the 7.5A factory for the original stereo)? Or should I use heavier wire?
Choose wire size based on load. Find out what the max power draw can be (in Watts). Then calculate amps based on max voltage. Us 15 V to be on the safe side. Most items draw more current when supplied at high voltage.
Choose fuses based on smallest and weakest wire or connection.

The underdash harness is in bad shape. The previous owner opted to bypass the fuseblock and not install any kind of fuse when installing the new stereo system. Lots of cable is just wrapped in electrical tape and ziptied there, and nothing is going through the fuse block.
Well maybe there's enough to salvage. Laying it out on the board will give you the lengths and let you replace the wires one by one. This assumes they haven't all been snipped. short. Since its running I'm hopeful it will at minimum be a guide.

Currently, only my ammeter gauge works. My fuel gauge is touch and go, seems fairly accurate, but not totally. Nothing from the temp gauge (it's not even hooked up on the back), and the lamps aren't connected.
The ammeter is not like other gages. It's big flat copper bar with a stud on each end. Current flowing through the bar moves the needle. Current flowing to the main splice moves the needle toward discharge. Current flowing into the battery moves it toward charge. When the battery is charged, there's no current flowing through it.

The rest of the gages work from a points operated voltage limiter. Clean everything, and check that the limiter's connections on th ecircuit board aren't broken. These gages work by heating a bimetal in the gage. Each gage is connected to the sending unit which is a variable resistor to ground.

Aside from that, just about everything is stock. I have a rebuilt alternator, not sure what the amperage output is from that, and just recently had to get a new alternator cable put in as the other one had burnt up.
That's bad but could be worse. Alternator output carries the highest current load of all the wires. When the engine is running it not only powers the usual stuff, but also recharges the battery. If the battery is low, that can add a lot of current at least for a few minutes. Stock alternators generally maxed out at low rpm which helped protect the wires a bit even when the battery was heavily discharged.
Current out depends on the what the demand is. What you could measure on a machine or with a Volt-Amp-testor on the car would be maximum potential output at a given rpm.

'64 should have an fusible link.
Fusible Links in Charging Systems with Ammeter

To make new terminal connections, you'll need Chrysler type connectors and an open barrel crimper.
Do a search here for that info.