Not specifically the route you are thinking of but a few suggestions. By 'southern Thai islands I presume you mean Phuket, Lanta, Phi Phi, Samui etc?? Why? That's a hell of a trek from Vietnam out of a tight-ish schedule.
The eastern islands offer a more relaxed pace and are not as plagued with full moon partygoing types as those in the south. Besides, you could make a whole holiday out of exploring the southern islands.
My suggestion would be Bangkok, Koh Chang (via Trat) then either on to Koh Kood (sometimes Koh Kut) or Koh Mak. There is a guy on Koh Chang that can arrange boats to Cambodia - your next point of call and you can then travel to Vietnam. The eastern islands of Thailand - Chang, Mak and Kood are my favourites and some of the beaches, especially on Koh Kood are stunning. They are less commercial (especially Koh Mak and Koh Kood) and offer various types of accommodation.
That sort of a route makes more sense geographically and you'd need to check this but a lot of the airlines offer one way tickets without penalty these days. You could therefore, possibly arrive in Bangkok and fly home from Vietnam.
What you suggest is do-able but I think the above is more practical and would give you more time on your actual holiday and less time in the air.
You can get to Trat by bus or plane. Buses from Ekamai station, Bangkok start at about 5.30am and take about 5.5 hours. There are different classes of bus available - not sure if the VIP type is available on that route but you would definitely want the express service. The last ferries from Trat to Koh Chang leave around 6.30 - 7.00pm so the earlier the bus you get, the better your chances of getting there the same day. Careful on this though as most buses drop you off on the main road near Trat where you take a Songthaew taxi (pick up truck) to the ferry terminal. Nothing wrong with that - just making you aware so you don't carry on to Trat City. There are also small minibuses available that will take you from Bangkok direct to the ferry terminal but I would warn against them - they are well known for accidents and dangerous driving. The bus is quite cheap. Get to the station 30 minutes before departure time and you won't have a problem getting a seat.
If you choose to fly, its an hour to Trat but its pricey. There is only one airline and they own the airport so they charge as much as the market will stand. There are minibuses at the airport that will take you the short distance to the ferry terminal.
The key to such a trip would be being able to get from Koh Chang to Cambodia (a spit away) and as I say, there's a guy on Koh Chang who claims to be able to offer such trips. I've never done it but I'm planning to next year.
Check that out - a lot of info on all the eastern islands and a specific tab on Cambodia.
One point, next to Christmas you are travelling at the highest point of the season for Thailand but search around and you'll still find some relative bargains.
Final suggestion, if you have to travel home from Bangkok - go straight to Trat when you arrive in Thailand (or do my whole suggestion in reverse). Most people do Bangkok first - I did but its a mistake. Things often don't go to plan and you end up rushing back to the airport at the end of your holiday - worse still, missing your flight home. Do the 2 days in Bangkok at the end - you're only 30 - 40 minutes from the airport. A bonus is also that if you've bought loads of gifts and souvenirs in Bangkok, you're not carting them around with you the whole holiday.