You probably don’t want to push too much power with a plastic reflector, so still keeping 4.5V from alkaleaks might work fine. I’d use a generic 7135-based driver. Dunno what size driver you might fit in there, but personally I prefer regulation across most of the battery-life, vs a constant decline and inconsistent light levels.

Hell, from FT you can get a 20mm star with 1 or 2 7135s on it, no separate driver needed! 700mA is pretty decent, and won’t cook/melt/char the reflector.