If there is a shortcut that makes it possible to skip this infeasible evolutionary process, I suspect that it would need to involve strong assumptions about common knowledge.

I've been thinking about this, and here are some possible shortcuts. Consider a strategy which I will call "Grumpy-Old-Man" or GOM-n. This behaves like TFT for the first n rounds, and then defects afterwards.

In your model, GOM-n for n = 150 would be able to drift into a population of TFT (since it has a very small fitness penalty of about 1 in 10 million, which is small enough to allow drift). If it did drift in and stabilize, there would then be selection pressure to slowly reduce the n, to GOM-149, then GOM-148 and so on.

Worse still, consider a mutant that produces GOM-150 but has selective advantages in earlier life, at a cost of crippling the TFT machinery in later life. (Not implausible because mutations often do have several effects). Then it could enter a population with a positive selective advantage, and clear the way for n to slowly reduce.

The particular example I was thinking of was a variant which is very good at detecting sneaky defection, disguised to look like co-operation. But then, as a side-effect in later life, everything starts to look like defection, so it becomes grumpy and stops co-operating itself. I know one or two folks like this...