Just run ZeroTier on the Mini and join it to your network. That's what we do. Of course we are ZeroTier so this is how we roll. Anything with an Internet connection is in "the cloud." We just donate all its spare cycles to Boinc. :)
Pay by the second would perhaps be preferable but unless I am reading this wrong you have to pay by the hour and it's relatively expensive.
Paying by the second is also ideal for these use cases. You run a build occasionally, but it mostly sits idle.