A much bigger problem is how your actions split the vote up. Say you run a "referendum ticket" with yourself and your left-wing Real President.
If the voter has two tickets: "you + lefty" vs. "righty", then you conflate left-wing vs whatever you're referending on - you can't know who supported the referendum, and who simply didn't want righty. So you say, "I know, I'll run another ticket with me + right-wing candidate," at which point you're now splitting the right-wing ticket (because votes don't magically sum), and the left-wing candidate will likely win regardless of support for your referendum.
So you introduce a 4th ticket, with lefty alone, so any voter can vote for left, right, left+you, or right+you. That sounds fair, right? Unless there's any correlation between party & support for your referendum. Let's say the left party prefers the referendum 80-20, and right party support for the referendum is about 50-50, and overall voter preference is 60-40 for right party. The winner of the ticket would be you+lefty (with 32% of the vote), even though 60% of people voted for righty.
TL;DR - first-past-the-pole only barely works with two candidates. You can't co-opt it to do a referendum.