Judging by the last one, you simply have a battery that is beyond help - it needs to be replaced.
With your car battery connected across, you can hear the pump prime (& stop) when you first turn on the ignition (this is normal on both counts); then you hear the starter
try to turn the motor, but there is insufficent current available.
I would take the battery out completely & put your track bike battery in there. Use the car battery with jumper cables also connected to it for additional cranking power,
You need to fix that bar switch - that is somewhat of a red herring, especially in the first movie - but absolutely need to open that up, make sure there is no possibility of a wire shorting to the bar and that nothing is broken.
There should be a pin on the switchgear that locates in a corresponding hole in the bar - the pin is clearly gone, Remove the screws from the switch, wrap electrical tape around the bar (inboard of the edges of the switch so it won't be seen) then re-insert the screws & tighten it down. The additional thickness of a few layers of tape will stop it rotating (and also stop the wires shorting against the bar). Do this BEFORE you go any further.
The first movie was a combination of completely dead battery and the switch making & breaking with insufficient power to complete the instrument test cycle.
With the indications in the third movie, there is no problem associated with clutch or sidestand switches - the kill maybe affecting thins but I don't think so.
Normal start switch operation is that the headlights will go out when you thumb the start switch.
When you said at end of movie 3 "I'm still hearing that same noise ...." to what are you referring?
Here is detail of the 'charging mod' -
http://www.triumphrat.net/daytona-de...-solution.html
If executed properly then that should not cause any issues to the problem you'e having. The PO appears to have connected it by my preferred method, directly to the battery and with an in-line fuse, so appears he may have done things correctly.
Make sure that the battery terminals are good and tight.
You could try putting your -ve jumper cable directly on the engine ground stud if you can reach it
If necessary, charge up your racebike battery overnight before you attempt to use it - or just go get a new one in the morning for the Triumph, it's going to need one anyway. Buy a GOOD one - spend a few extra dollars for a quality one.
After you get it running, we can check on the general condition of the charging system. Regulators are notorious for failure on these bikes.