Several things here:
Allow base coat to dry longer (24 hours or more). The surface of the Mona Lisa red basecoat will dry to touch in several minutes, but the trapped solvent (water in this case) will take much longer to eventually evaporate. Balsa is an absorbent wood with a stringy texture and can hold a lot of water, adding to the length of time needed for curing.
Water based size cannot be burnished. It will always be a somewhat matte surface. This is a different process than traditional water-based gilding methods or the Kolner Water Gilding System. Water based size will form a tacky surface, essentially glueing the composite leaf to the surface. It is convenient for craft projects, but once the leaf is down that's what you are going to get. Slow setting oil-based size is recommended for higher quality results and can be burnished (to some extent with composite leaf).
It will tarnish, even inside. You may want to varnish with an oil-based varnish that has UV protection formulated in the product (it will have a light amber tint). Traditional spar varnish works well, but takes forever to dry and may need some kind of protection from dust and bugs while it is drying. If good enough is good enough polyurethane or acrylic varnish can be used.
22k "SignGold" vinyl is available in rolls and can be cut on a plotter. It has a UV barrier top film and can be cut and applied just like regular
sign vinyl. It is expensive, but you will make it up in time.