Had the same problem - solution was to chop a bit of the bottom of the door off. Unless you're completely OCD then who gives a monkeys if it looks a bit "bespoke"?
Seen that a few times , a door notched to go around an obstruction and I seem to recall seeing on the internet ( it might even have been linked from this site) a door that had been cut to pass a wash hand basin
So it looks like it's back to your original idea of a piece of ply or perhaps thin mdf depending on the finish you are applying.
Yea I think so... The final finish will be Dulux satinwood so think the plywood should be ok. I will have a go tonight!