Insulate small suspended ceiling
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I think I would blow in insulation to fill the cavity, and then get one of those magnetic vent covers that you can remove when showering. https://www.amazon.com/s/ref=nb_sb_ss_i_4_19?url=search-alias%3Daps&field-keywords=magnetic+vent+covers+for+ceiling&sprefix=magnetic+vent+cover%2Caps%2C211&crid=1L1M9BEB9JO77
Or you could get some 1" rigid foam insulation, and slide panels of it in above your ceiling tiles. https://www.homedepot.com/p/Foamular-150-1-in-x-4-ft-x-8-ft-R-5-Scored-Square-Edge-Insulating-Sheathing-20WE/207179253?keyword=foamular+rigid+insulation
blow-in.... https://www.homedepot.com/b/Building-Materials-Insulation-Blow-in-Insulation/N-5yc1vZbayp?NCNI-5&searchRedirect=blow-in+insulation&semanticToken=20082+++>++++st%3A%7Bblow-in+insulation%7D%3Ast++cn%3A%7B0%3A0%7D++blow-in+insulation+%7Bcategory%7D
If you have access to attic/crawlspace elsewhere, you could also seal off the area with plywood and put insulation on top of it.
Some different type of closing vents.... https://www.google.com/search?q=self-closing+bathroom+vent&client=opera&hs=62S&tbm=isch&tbo=u&source=univ&sa=X&ved=0ahUKEwjqqqK98cPXAhVS-GMKHUlRCmAQ7AkImQE&biw=1366&bih=629
Can you insulate above it by using access to crawl space? If so, install the pink fiberglass insulation covered by foil. That will insulate some. If not, you would probably need to remove the ceiling and insulate from below. Then re-install the ceiling. Get the highest R factor when purchasing your insulation. There is also insulation you can have blown in, but again, you need access to the space above the drop ceiling.
I can't answer with regard to the vent, though.
What type of insulation is in the attic? You may not have enough. Yes, there are many vent options. See if this link helps.
http://www.hgtv.com/remodel/bathroom-remodel/choosing-a-bath-ventilation-fan
There are fans that have a trap door that opens by the fan sucking the air up but you might just want to not use the fan in the winter and cover the entire area above the 12 x 12 with a couple of layers of roll insulation
Also, get a heated floor pad and put carpet over it. That will keep the area warm. https://cozywinters.com/shop/rug-heat.html