Finally got the problem solved! I think we had multiple problems.
At first, the small brass inserts in the crossbar of the air horn were blocking the fuel spray. Fixed that.
Then we had an issue with our needle and seat.
The rebuild video from Mike's carbs advocates spraying some air in while you gently hold down the needle to make sure it seat well and doesn't allow air flow. Our original seat sealed better, so we went with that, and left the new one in the bag.
One of the classic car guys in our town came by today and looked at the carb. He showed us that the tab on the float wasn't pushing the needle into the seat in a uniform manner. It should push the needle directly up, or as nearly so as possible, and ours was pushing the needle up at an angle.
So, after rebending the tab on the float to make it push the needle in a straighter fashion, getting the float adjusted properly (1 9/32" with 1 3/4" drop), and then replacing the seat with the one from the rebuild kit (despite it not seeming to seal as well), everything runs now.
It may still be a bit rich, but we're not dumping fuel everywhere now, and the truck runs smoothly (much better than before the rebuild) and we're in business.
Thanks to everyone here who chimed in!
