I don't know, if I had the details and felt confident that the person was indeed guilty, I don't think it would cause me much anguish to push the button. However, if I didn't agree with the jury's decision of guilt, forcing that person to do it would be extremely inhumane, it could ruin your life.
In fact, to go a bit further with this—imaging that ongoing punishment required getting this group of people back together at regular intervals and getting them all to push the button again. (For example, meting out a lifetime prison sentence one year, or one month, at a time.) It isn't meant to cause re-evaluation due to the meeting being a hassle, so maybe they could just do this online—but it should cause re-evaluation of whether each juror still feels as strongly about the person's sentence now as they did before.
One could even imagine a situation where any capitol punishment would be measured in QALYs, and then the jury each have to do that many man-years of confirming in order to apply the punishment. (Thus, applying a sentence of death to an eighteen-year-old would require decades of total work, or years per juror.) That seems about the right weight upon one's conscience. It would also mean you'd probably never be able to do jury duty more than once in your life.
I'm curious about your argument. The death penalty was suspended, limited, limited again, and finally the last loopholes allowing it in special case were closed by the passing of more than one bill by parliament, none of which were referendums. Are you basing this on polls?
Reminds me of various cables from the front sent to commanders of the 'Axis' where they lamented how soldiers were uncomfortable or downright refusing to execute prisoners so they abstracted the entire nasty work of organized state murder away to solve this problem. Replace human conscience with bland prison bureaucracy and systems structures. Everybody passes paper, nobody thinks they are ethically responsible.