Can electrons be electrically polarized, i.e., can they acquire an induced electric dipole moment?

The neutron electric dipole moment (nEDM) is a measure for the distribution of positive and negative charge inside the neutron. A finite electric dipole moment can only exist if the centers of the negative and positive charge distribution inside the particle do not coincide.

You say :

For the electron, a nonzero electric dipole moment can be understood (in an inaccurate classical model) as a spatial separation between the center of mass and the center of charge.)

Within the standard model of particle physics, electrons are point particles, they are not composite as the neutron is ( full of quarks and antiquarks ). The center of mass and the center of charge are by definition the same. No distribution of charge can exist at a point, by definition of "distribution", so no classical definition of an electron EDM can be envisaged.

As far as I understand it, the experiments trying to measure the intrinsic electron EDM are checking calculations of the standard model, where feynman diagrams with various exchange loops can be considered as defining a probabilistic quantum mechanical charge distribution for the interactions of the electron. These are very very improbable as the numbers show.

Thus your question involves theories beyond the standard model as for example preons, where quarks and leptons are considered composite, no longer point particles and a spatial charge distribution of the orbitals of the preons could exist, similar to the quarks-antiquarks within the neutron.

In this link

It is well known that the electron has a magnetic dipole moment, which is a result of the particle’s “spin”, or intrinsic angular momentum. However, time reversal symmetry – the requirement that physics is the same for time running forwards and backwards – forbids the electron from also having an EDM.

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Time reversal symmetry is a tenet of the simplest version of the Standard Model, so any measurement of the EDM would point to new physics. Some versions of the Standard Model do allow some violation of time reversal, but this would result in an EDM smaller than about $10^{−39}$ e cm. This would be extremely difficult to measure experimentally. However, some models that attempt to describe physics beyond the Standard Model predict much larger EDMs for the electron, and these predictions can be tested in the lab.

The link goes on to describe experiments

If time reversal symmetry is violated it leads to CP violation so the discovery of an electron EDM , not limits, will signal new physics beyond the standard model, as studied here .

In the comments you state:

That said, if you know of a strong proof that the fact that the intrinsic eEDM comes via weak interactions means that induced eEDMs are impossible, then by all means, link to it

In my opinion, induced eEDMs are impossible in a model where the electron is a point particle. After all the intrinsic EDMs calculated from particular feynman diagrams are a tyoe of induced EDM, except in the quantum mechanicaly probability space. If no intrinsic electron EDM is firmly established experimentally, there is no interest for theorists to involve themselves in compositeness theories , where the electron is no longer an elementary point particle.