The support bods are right in that you can't run an in place upgrade from 32-bit to 64-bit. Depends on how the environment is configured as to how much ballache it'll be re-installing Windows then the apps and all the settings that come with it manually.
I can kind of see the point they're making. You're looking at £120+ on a Windows license....
You can use the existing license for moving from 32 to 64x.
We have the same issue where around 400 PCs are on 32 bit and it's a right pain for my GP's especially folder redirection's etc.
Would you be allowed to re-image the machines yourself? I would reply to that support email, it's insane as i assume your company pay a support fee annually. Even if the work takes x amount of days, this shouldn't factor in as they're paid to do this.