I'd say readdress the basics. I'm a little confused because because at the beginning your post says that it won't even catch and then later you say it belches black smoke.
Make sure that you are in fact getting fuel. Turn the petcock to prime to eliminate the vacuum petcock. Prime overrides the vacuum diaphragm. (Also make sure that the petcock knob is actually turning the shaft, that the knob is not broken and spinning on the shaft. I'm on my 3rd knob)
Make sure that the battery is fully charged.
Replace the spark plugs, they sound like they're fouled.
Do all three of these at the same time, maybe it will fire and you can get more information from how the bike is running to see what else is wrong.
When you rebuilt the carburetors did you replace the float needle and seats? They become a problem in a bike of this age.
Another common failure is the crank position sensor (or pick-up coil) fails but it usually quits when the bike is hot and not a cold engine failure, emphasis on "usually".
You might consider moving this post to the T-3/sport touring forum as the Trident owner's don't always come over here to read.
http://www.triumphrat.net/t3-sport-touring-forum/
