In theory, you can. You can build the hot end of Stirling engine into part of the 2nd loop as the heat exchanger. I have heard such proposal.
In practice, maybe nobody want it for reasons:
- Stirling engine use permanently gaseous working fluid. It is worse than liquid in transporting heat energy, meaning lower overall efficiency.
- Stirling engine is piston engine running at much lower RPM. Lower frequency of noise travels longer range than high frequency noise of the steam turbine.
- Stirling engine's pistons moves in a linear vector back and forth. It create strong vibrations, the bigger the piston is, the stronger, this is related to 2nd point, but not limited to noise.
- Stirling engine is big and bulky, lower power density compared to steam turbine.
- Practical Stirling engine has pumps in the loop. The working gas need to be cooled by ambient air or sea water in a sub. That circulation need pumps. It does not eliminate pumps more than this nuclear AIP who only has pump in the 3rd loop, the 1st and 2nd loop are natural circulated.
- Stirling engine has much lower efficiency than steam turbine. The nuclear AIP's reactor has already sacrificed efficiency for noise, size and safety reasons, using Stirling engine will make it useless.
There is nothing bad with steam or turbine although they sounds old because they were broadly used for a long time. Stirling engine is nothing new, nor technically superior.