In the first problem, the agent could commit to one-boxing (through the mechanism I described in the link) and then finish simulating the predictor afterwards. Then the predictor would still be able to simulate the agent until it commits to one-boxing, and then prove that the agent will one-box no matter what it computes after that.
The second version of the problem seems more likely to cause problems, but it might work for the agent to restrict itself to not using the information it pre-computed for the purposes of modeling the predictor (even though it has to use that information for understanding the problem). If predictor is capable of verifying or assuming that the agent will correctly simulate it, it could skip the impossible step of fully simulating the agent fully simulating it, and just simulate the agent on the decrypted problem.
In the first problem, the agent could commit to one-boxing (through the mechanism I described in the link) and then finish simulating the predictor afterwards. Then the predictor would still be able to simulate the agent until it commits to one-boxing, and then prove that the agent will one-box no matter what it computes after that.
The second version of the problem seems more likely to cause problems, but it might work for the agent to restrict itself to not using the information it pre-computed for the purposes of modeling the predictor (even though it has to use that information for understanding the problem). If predictor is capable of verifying or assuming that the agent will correctly simulate it, it could skip the impossible step of fully simulating the agent fully simulating it, and just simulate the agent on the decrypted problem.