I don't think your "detect infinite resources and cheat" strategy is really worth thinking about. Instead of strategies like CDT and TDT whose applicability to limited compute resources is unclear, suppose you have an anytime strategy X, which you can halt at any time and get a decision. Then there's really a family of algorithms X-t, where t is the time you're going to give it to run. In this case, if you are X-t, we can consider the situation where Omega fields X-t against you.