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Uncle Beer, you´re correct, of course.

In order to take care of that, you don´t lean the horn against a wall, but into a corner of the shower booth.
That way, the horn has sufficient resting points to stabilise its position:
- bottom bow rests on the ground and won´t slide because of the friction provided by the wet towel.
When in doubt, you might want to block any sliding motion with suitable objects.
This defines vertical position, but offers no stability against rotation or toppling over.
- bell rests against the corner, defines upright position, but can´t prevent the horn from rotating around an axis originating from the bottom bow´s resting point and running through the center of the bell´s rim.
- horn should be rotated so the main pipe between top bow and side branch will also rest against one of the two walls that make up the corner, which takes care of that last remaining freedom of mobility.

Any increase in weight, no matter how it may be distributed inside the horn, will have its center of gravity within the base created by those contact points between horn and structure.

Here, where we have lots of horses, I would go to the "pet" store and get a horse water trough big enough for a tuba, and use that. It can't fall over if it's already lying down.