How can I modify the size of the start marker independently of the width of
the segment?
I was going to suggest unchecking the 'scale stroke width' option under 'Inkscape Preferences->Transforms'. but I tried it and this will let you scale up the line with arrow head but the line thickness AND arrowhead both don't get scaled with this option.
I guess the workaround might be to create a short line with the arrow marker. Drag the end node into, or close to the arrowhead and then scale it up to the size you want. Then of course use a separate line for the line portion, then group them together. A definite kludgy solution, but it works.