Actually, Alphaseadog, the advice still does apply. A lot will depend on what time your flights arrive and depart from Bangkok and the kind of diving you want to do.
Since you want to tack diving on to the beginning or the end of your trip, if your ticket takes you to Bangkok in good time to get to Phuket or Samui on the same day, you could possibly start diving the next morning. For example, let's say that you are on a flight from the west coast of the US that arrives in the evening in Bangkok. You can still get a flight out to Phuket definitely, and possibly to Samui. When you arrive at Phuket, you are there and only need to go to your hotel. When you arrive at Samui, you can either dive with operators from that island (the diving is generally at Koh Tao), or you can take a ferry to Tao and dive with operators based there. Ferries schedules may make it more convenient to use Samui-based operators. There is also the possibility of driving overland to Pattaya and diving from there. The attraction of Pattaya-based diving is shipwrecks. If you like them, that's the place to go. If you can take them or leave them, go someplace else where the marine life is more plentiful and the visibility is better.
So let's say that you choose to fly to your dive destination (Phuket or Samui/Tao), and that you dive the day after you arrive plus a second day. Whether you can dive on the third day will depend on when you need to be in Bangkok to work or to catch your flight out. Our dives tend to end in mid-afternoon, so if your flight is an afternoon flight, you might be able to squeeze in another dive day, but if you have to get out in the morning, you'll have a free-and-easy day with no diving.
The question of whether to come to the Andaman side or go to the Gulf side may depend on whether rain makes any difference to you. September is statistically a very rainy month on Phuket (Andaman side) and somewhat less rainy in the Gulf. This website has an easy-to-use map showing weather statistics. In addition, two Thailand forum members active here blog and include weather observations: Camille and Jamie.
Finally, a note on rain: our rain is warm, so feeling frozen on a boat in the rain after a dive isn't an issue. And we don't have a real problem with runoff, so the rain does't diminish visibility to any great extent. But what rainy weather in the Andaman does do is make the boat rides a bit bumpier, and can also make swimming off our beaches inadvisable due to rip currents. However, the fish don't care at all about the rain, so the actual diving is mostly unaffected, except that the boats are not as crowded, meaning the dive sites are not as crowded, meaning that animals like leopard sharks are less likely to swim off to find peace and quiet somewhere that we don't dive. I think diving from Phuket during our low season is quite pleasant!
