Just 5 days

There is no good diving near Bangkok. The closest place would be the south of Ko Chang, but it takes 6+ hours by bus/boat to get there from BKK. There is some diving around Pattaya a few hours from BKK but it is generally not good. Ko Tao takes a whole day to reach from Bangkok by bus/boat, and still takes most of a day if you fly to Ko Samui and then take a boat to Ko Tao. Try Ko Chang or forget it. I'm just jealous you are going to Bhutan - always wanted to see that country but $200/day is way steep.

5 days in Thailand good enough to go diving. I would go to the domestic air carriers once I arrive at the airport and ask for the next available flight to an island in the south and go straight from there, no need to go into town, and still very cheap. All the flights are about 1 hr long, give or take 15-30 minutes so nothing unbearable. What I did was the same thing, just showed up at the airport and booked the next flight to Krabi, worked out perfectly. You just need to do your homework and pick out where to go, that might be the hard part since there's so many great places to dive. There's nothing good around Bangkok, esp for diving so just go to the islands, you'll be glad you got away from the city. If you're a planner, you can book it ahead but no need, it's really easy to do it on the spot in Thailand.