No I dont think it would, but I have heard of retrofitters using pc ballasts to overrun NO bulbs (why this is done I have no idea but it is electrically possible).
Considering the cheapness of 13w ballasts and bulbs, Id feel better recommending the complete 13w setup either as an azoo galaxy light or the packs offered by hellolights or that other link that was posted. This seems safer in terms of avoiding fire hazards and also getting the most longevity out of the aftermarket bulbs you will use in the final design.
That is really good sealing, you lid fits on all sides. Did you decide to go with the 70/30 option, the one where a small pwrhd does %70 of the circulation work and where a small airstone would output %30 of circulation, mainly to keep fresh air under the lid? I think that will work best.
Also, you may not even need an airstone. If not, you will have the very least systemic evaporation. If you leave a good two inch gap under your sealed lid, this likely collects enough air to provide balance even through the night cycle. A new problem would be CO2 accumulation but this is workable if the tank has a small relative bioload.
This again would depend on active fish load... each time you raised the lid to dose (often if you have stonies) air would refresh under the lid providing enough oxygen stay for the next go-round. There is not that much demand for oxygen past what standard circulation will provide if your nano reef uses little or no fish.
I still think you could keep a small fish in there with no problems after the tank has been fed and cycled for awhile.
Brandon