While there could be various approaches building studless the easiest one would be with Tile, Modified 2 x 2 Inverted. These modified tiles have studs on one side, which fit under bricks and plates, and smooth surface on the other side.
Here is a prototype which features equal 6-stud (or 30 modules) long sides. Parts can be changed upon availability, but it ...
According to the official description, these are for customizing the completed build:
This LEGO Batman toy playset includes 2 minifigures and children can customize their builds, minifigures and weapons with the included pack of 10 assorted Bat elements.
As mentioned in another answer, there are dedicated pieces for this now.
But... 6x6x6 cube is plenty big enough to have a stud-reversing construction inside, then the bottom can be a simple 6x6 plate with regular smooth tiles.
For example put some of these inside the cube:
There doesn't seem to be a difference in a shape of triangles (that's how they mention them in the video) seen in Dark Azure as well as Dark Blue colors. However there are two shapes - bigger and smaller triangles. Smaller triangle's edge is half the length of bigger one - you can attach two of these to bigger triangle.
Larger one requires parts listed ...
It actually does take a Technic axle and has been used with such in some sets, although in most old sets I know it was usually supposed to be operated manually by rolling the drum with your fingers.
If you look at the picture of the individual drum piece, you can just about see the axle holder inside, even though the round hole itself is bigger. So you can ...
This isn't my personal experience, but I have seen and heard people use rubber hammers for fixing larger plates onto baseplates, which shouldn't be all too different from fixing them onto normal plates.
Those are a classic homeworking tool, commonly used pretty much for the "real world" equivalent of what you're doing, fixing floor tiles and cobblestones ...
Your question sparked an idea of making a completely symmetrical 6x6x6 cube with the same tile on each face.
This is what I came up with:
The LDCad file is here.
6x-10202, 8x-26604, 8x-30414, 2x-3958
I used only parts available in LDCad version 1.6b
I had a whole bunch of pieces that were flat, 2x2 with studs on both sides. They were not actually LEGOs but some knock-off brand - I want to say Tyco? Not sure, it's been a while, but it was 100% compatible. I used them for this sort of thing all the time. It annoyed me that there weren't any official LEGO pieces that served this purpose at the time, ...
I'm very late to the game here, but why can't we use the existing hole as a guide for an appropriately sized drill bit, converting it from Friends accessory size to standard LEGO horn size? If you drilled it slowly by hand, it would be really hard to mess it up.
I would suggest using part 15712 'tile 1x1 with C clip'.Just place that on top of the minifig head and then place the white part in the clip.
You could have used a bar with clip but most if not all minifig heads dont have a fully hollow stud, only solid, or semi-hollow, so the bar would not be able to go all the way down and even if it did the minifigs torso ...