You don't convert your display LUT.
You generate a Rec709 to Rec601 transform LUT, and apply that to the entire project for the Rec601 deliverable.
You also generate a Rec709 to DCI XYZ transform LUT, and again apply to the whole project for the DCI deliverable.
We do this all the time, with no errors at all.
The converted colour space versions will match the original Rec709 version when displayed on accurate Rec601 and XYZ displays.
There will be nothing to correct, as there will be no errors. The conversion will be 'clean' and accurate.
And you wont find a cheap display that can do P3 accurately - they don't exist.
Mob Boss at Light Illusion