How can I export drawing from Notes to an image with background on an iPad?
A third, more crude method, which you can use if a certain part of your doodle requires background lines/grids is taking  screenshot of the doodle.

Step 1;
Expand the note using the button in the top right of the screen.

Step 2;
Hide the panel showing pens and colors using the “x” at the bottom right of the panel.

Step 3;
Take a screenshot by pressing the home and power buttons at the same time.

Step 4; Crop the screenshot to show only the desired result.

The end result is something like this;