Printing the mushroom:

I printed with a layer height of 0.16 in vase mode. 

On the bottom of the print  there is a 0.6 mm deep rectangle valley so for example in Cura slicer if you uncheck the “Remove All Holes” setting and set the number of bottom layer to 3 then the mushroom vase will have the rectangular hole you need. After finishing the bottom layers it will continue with the vase mode as normal.

Printing the support:

I printed with 0.2 layer height, no support. NOT in vase mode.

Assembly:

Two screws in the wall to hold the support. Then as you insert the mushroom you should be able to push it down a bit so the bottom layer of the mushroom will get in-between the wall and the support piece.