Ditch the tab navigator.
Use a tabBar to toggle the children of a viewstack, put that tab bar inside a box, set the boxes to a specific width, then the box will give you a scroll bar when the tabs extend beyond the width. You can move/style that scrollbar so you can line everything up right.
By separating the tabBar from the navigator (in this case the view stack) you can put it in a different parent and get the result you're looking for.
As per your comment:
I don't have an example but there is something similar here:
http://fleksray.org/adobe_flex_components_en.html#Button%20Scrolling%20Canavas
This is how I would do it:
/--------------------------------------------------------------------------------------\
| BOX-A |
| /--------\/-----------------------------------------------------------\/---------\ |
| | || || | |
| | BOX-B || BOX-C || BOX-D | |
| | || || | |
| \--------/\-----------------------------------------------------------/\---------/ |
| |
\--------------------------------------------------------------------------------------/
Box B is just big enough for your arrow image, same with box D. Box C contains your tabBar, but turn horizontalscrollpolicy to off. All 3 of these go into Box A which can be a HBox to make layout easier. Add a click event handler to your arrow images to set Box-c's horizontalscrollposition +=20, or -=20 depending on your arrow. (change those numbers to suit you). You can even go the extra mile and set the includeInLayout properties of boxes B and D to false unless your tabBar takes up more than the width of Box C.