Connect two segments without affecting the selected profile option (stroke panel)

Charlotte,

"I would like to know if it is possible to join these two segments on Illustrator, but while keeping the "profile option" selected (on strokes panel)."

The long answer is no.

But if the appearance of those two being joined is sufficient, it is possible in multiple ways, and you may choose the least worst one.

To mention a few, you may:

A) To keep a stroked path you can edit freely: Create a closed path that can cover the unwanted end (you may copy a segment of the crossing path to start with), then give it a black fill if not already, then select both paths and in the Transparnecy palette (flyout) click Make (Opacity) Mask with both Clip and Inverse Mask unticked (in the main Transparency window).

B) Select the stroked spiky path, then Object>Path>Outline Stroke, then cut the outlined path where it crosses the thicker path, then delete the unwanted spike part, and with the wanted part selected Ctrl/Cmd+J to join the ends and close it.