But if we view this as an actual (albeit unrealistic/highly theoretical) situation

There is no such thing as an actual unrealistic situation.

A perfectly rational agent can realize that the problem has no optimal solution and mark it as unsolvable, but afterwards they still have to pick a number

They do not have to pick a number, because the situation is not real. To say "but suppose it was" is only to repeat the original hypothetical question that the agent has declared unsolved. If we stipulate that the agent is so logically omniscient as to never need to abandon a problem as unsolved, that does not tell us, who are not omniscient, what that hypothetical agent's hypothetical choice in that hypothetical situation would be.

The whole problem seems to me on a level with "can God make a weight so heavy he can't lift it?"