I'm going with "it's too dangerous, so don't do it". I have two manufactured alcohol stoves, and both contain warnings against refill while hot.
That being said, my answer is "You don't".
Pictured (I can't see the video) suggests a squeeze bottle with a plastic tube running into a reservoir where the alcohol is supposed to be while flames are clearly showing.
My first concern would be the squeeze bottle. Although a flame requires heat, fuel, and air, a flame is not likely to go up through the tube and into the squeeze bottle, ready to ignite, since there is no air in the tube, and cold(er) fuel is being pushed out of the tube. Also, liquid fuel doesn't really burn - its vapors burn. So, there's nothing flammable in the tube.
Or is there?
What happens at the moment you decide you've added enough fuel?
You can keep the same squeeze pressure so that there is zero net pressure on the squeeze bottle: no positive pressure which would push more alcohol out of the tube; or negative pressure which would suck in alcohol from the stove or flaming vapors abouve the fuel line. Probably, nothing would happen, except the end of the tube - having passed through the flames - would also ignite. What's your plan to put this out safely?
You can apply positive pressure on the squeeze bottle, ensuring nothing gets sucked back; but now you have a stream of flammable (and flaming) fuel at the end of the squeeze bottle. Also, if you apply too much positive pressure on the bottle, you can potentially create bubbling in the reservoir of alcohol, and those bubbles could exit the stove creating a severe fire hazard around you.
You can also apply negative pressure on the squeeze bottle, but this seems more dangerous: the outside of the tube has flames, and now, you're sucking in vapor and flames into the tube and, subsequently, the bottle. Potentially more dangerous is supplying too much negative pressure, and siphoning all the (hot) alcohol from the reservoir back into the bottle.
There doesn't seem to be any safe way out of this, as all options lead to a flaming tip. Even if you decided one method was safer than the others, there's no guarantee you could maintain that every time, nor could you guarantee everyone could do it. Perhaps, you have a specialized steel filter placed inside the tube; this absorbs heat and prevents blow-back into the bottle with errant negative squeeze. Nevertheless, you have a tip of flaming alcohol at the end of the tube, and this is inherently dangerous.
Bottom line is that with any method you choose to fill up, per the photo, it would require delicate dexterity to maintain the precise amount of pressure on the bottle to not cause accidents.
One possibility is to have an external reservoir available outside the stove proper. A sort of "L" pipe which leads to the bottom of the reservoir, and the other end with a funnel shaped end leads sufficiently away from the stove apparatus as to not allow the vapors to ignite, and allows to pour fuel into the funnel. The problem with this is that in pouring, you create more vapors that could be ignited; and if the stove were nearly out of fuel (instead of just a little bit out of fuel), the vapors in the tube and funnel are now flammable.
So an external pipe assembly does not make much sense. Besides, it's probably going to be cumbersome to use.
All of this work - and danger - seems overpriced compared to a simpler solution: put the stove out, refill, and relight. It's not like a pressurized canister, so it's should be easy to accommodate.