Transparency

If you want to make your swallowed FvwmPager transparent, you have to make the FvwmButtons transparent, too.

Pixmap none

will not work, you have to use:

Pixmap /path/to/transparent/image

for both modules.

But it is much better to always use colorsets. So, instead of Pixmap, use Colorset <n> Transparent/RootTransparent, and then Style FvwmButtons ParentalRelativity.