It took me a few minutes of fiddling around to work out the problem, but when I did it was a bit of a face/palm moment.
The project I was working on had some customisations in Microsoft Dynamics GP as well as a new integration template. When I logged in first thing I had started GP using Run as Administrator and then launched Integration Manager from the Tools menu (); but, when reopening it, I had launched it from the Windows Start screen.
This meant that GP itself was running under the administrator account, but Integration Manager was running under my user account. I closed Integration Manager again and then launched it from the GP menu, so that it was running under the same user account as GP itself, and was able to run the integration without further problems.