That's a tough question to answer. Besides the choke the difference is calibration. The 1406 is more of an emission carb while the 1405 is more of a performance and richer.
The only source of switched 12V is off the ignition switch to the coil. You'd have to T the power at that spot for the choke. I would suggest running a relay as well. The choke will be more of a drain then intended. As a result the choke may not work too well as well as the ignition.
I have a 1406 on my 66 as well. I'm sorting it out after sitting a number of years. I bought the carb used. It was jetted by the PO to the next step richer, 1%. I returned the carb to stock calibrations and found it way too lean for stock. Had a terrible stumble and bog right off idle. I went back to the 1% increase and it runs much better but I want to fool around with it more once the snow is gone. Right now I have the choke disconnected for the reason I mentioned after thinking about it. I haven't put a relay in yet but I'm going to. I also bought a used Speedway GM style HEI distributer and I'm going to run a relay on that as well. Think about it. These draw a lot more power then was designed 40+ years ago for just a simple ignition coil. I'm sure both of these running off a relay will operate much, much better!
If you buy the 1406, you may have to buy the calibration kit too. The booklet supplied with the carb is excellent. It describes how a carb works, has calibration charts telling you how much in percentage and where it make the changes in operation.