Adrian, not sure if this problem is still relevant to you 2 years later (amazing that this is still not customizable after all this time!), but after some tinkering I found a workaround that will hopefully work for you:
1. Use Chrome (other browsers should work but I only know how the console works in Chrome)
2. Create a link back to your survey in the text editor as you normally would.
3. After everything is configured exactly how you want it, highlight your link, right click, and click "Inspect". This will open the console on the righthand side.
4. If you scroll up or down a bit in the console (make sure you are on the "Elements" tab), you should find something that says "target="_blank"". Replace the word "blank" with the word "self".
5. Click save and publish your survey as you normally would.
Something to note, I believe this will reset itself every time you edit the thank you screen and possibly every time you edit the survey in general. So if you do this to a survey you edit a lot, I would test this link a lot too to make sure it's still working.
I really hope this helps and I hope ESRI makes the "submit another response" option customizable!