How to twist a complex shape in Inkscape?

I have a shape derived from the letter i:

Each of the blue “waves” and the two orange crosslinks are a path, the dot is a circle.

I want to make the shape derived from a script i instead, following roughly a curve like this one below the dot:

I tried using Pattern along path, but it ended up doing strange distortions and not what I imagined:

It is OK if it can’t be done at once, but I need to apply the same effect to all of the four paths in my shape, or maybe only to the two blue paths – I can somehow arrange the orange paths afterwards. But the blue paths are complicated, with many points, and I don’t think I will get a good effect if I try to bend them by hand.

Any ideas how to do it, if it’s possible at all?

If you want to play with the SVG original, I uploaded the source on pastebin.

This is not the same question as this one. I assume that it is because of my complex shape instead of a simple rectangle – the solutions there don’t work for me.

Update Wrzlprmft's method produced this when used with a two-loop segment traced with Bezier curves:

Beside the strange squish, I'm still looking for a way to use it with an existing combination of curves, instead of having to redraw them using other methods.