Wiring an alternator output to the battery side of the starter relay, is essentially the same as wiring it directly to the battery.
The only thing "extra" is a few inches (usually) of additional battery cable. If all of the cables are in good condition, then there is not enough resistance in that short run to make a difference.
Again, essentially the same thing as connecting it directly to the battery.
The fuse is always a good idear...
As the others have said as well, if your battery can crank the starter and start the engine, then it should run fine with, or without the extra couple of volts of the alternator charging.
But if you have a computer controlling anything, that is expecting to see full voltage all the time, and better yet, the 14-15 volts of a working alternator, that could explain why it's acting up.
With all the new wiring, I would certainly not rely on one of those "temporary patch connectors" like you are using on your battery terminal. Yes, they can work fine, and do so for quite awhile. But they have too many down sides to not try something a little more on the permanent side.
If you said you had a carb and points ignition, I missed it. Since you have a Pertronix coil, do you have a Pertronix Ignitor distributor, or similar?
The resistor that we talk about all the time is not visible on a Bronco. It's the resistor wire between the ignition switch and firewall, which turns into the Red w/green stripe wire that runs to the coil.
If you are running the original wire to the positive side of the coil, and running the Ignitor off of that, it could be at least part of your issue.
There are plenty of documented cases on the forums over the years of engines that ran just fine initially, but at some point weeks, months, and even years later started giving trouble when it was running with the original resistor wire.
What exactly are the wires on your coil? Is that a Red wire from the distributor on the positive side, and the Black wire from the distributor on the negative? What's the additional wire on the positive? The original Red w/green?
You can test this theory, or of any issue with the ignition switch too probably, by temporarily running a jumper wire from the battery positive to the coil positive.
Remember, do NOT leave the wire connected. Only connect it when you're ready to fire up the engine, then disconnect it to either stop the engine, or in the case of the engine stopping on it's own.
Just like leaving the key in the ON position, never leave a hot wire connected to the ignition coil.
Paul