There's a relatively straightforward trick that you can do here. If you've got a transform that turns a circle into an ellipse with the relative radii and orientation that you want, then:

1. Apply the inverse of that transform to your path.

2. Stroke the path.

3. Apply the transform to the result.

This way, the path stays in place but the stroke is transformed to give it a calligraphic look.

With a JavaScript Canvas, you can just set the transform after defining the path, but before stroking. JSFiddle example: [0].

(This was something that I tested in my tiny, single-header <canvas>-like 2D rasterizer library for C++ and my Javascript port of its test suite [1].)

For Inkscape, I think you can convert an object to a path, apply the inverse transform, do a minimal simplification to bake the transform into the path, stroke it, and then apply the forward transform. It's a bit clumsy, but I bet someone could easily create an extension script to do it.

[0] https://jsfiddle.net/y7m16wa0/

[1] https://github.com/a-e-k/canvas_ity/blob/main/test/test.cpp#..., https://github.com/a-e-k/canvas_ity/blob/main/test/test.html...