How to raise this floor transition joint?
You need to take into account three issues: Pressure from the top (someone standing on the strip), lateral pressure (someone knocking or stubbing the trip forwards or backwards), and moisture (what goes under the trim when mopping the floor).

To address each of these issues:


Place support under the flat surface of the trim
Ensure some of the bracket is still available, and 
Use wood that is tolerant to moisture


I had a similar problem (with an identical piece of trim) and this is how I fixed the issue.  It's not perfect, but you'll hardly notice!


Use a chisel to remove the small rectangle of wood that the strip connects to. (The X in the picture)
Cut the floor strip to remove the right side of the plastic. In the picture below, it would result in the plastic strip looking like an uppercase letter L.
Dry install the strip and the transition.  You may find the strip is too tall (i.e. the 'back' of the L) to let the transition sit flush with the floor. If so, shave down the strip.  Be careful, these guys shatter easily. A file may be best.
Align and install your strip.
Find a piece of wood (perhaps a water resistant MDF) that is the right height to place under the transition.
Install the spacing wood. Personally I would screw it to the floor. 
Apply construction adhesive to strip and top of the spacer. Your connecting  strip is compromised - be generous with the adhesive.
Leave to dry, best of luck with your project!