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I think there's two more in the rrule(::typeof(⊗), ...) in ext/TensorKitChainRulesCoreExt/linalg.jl. I assume these will have the same issue, so might be good to change these here too?

Unfortunately, the non-passing tests seem to be unrelated. I still haven't figured out a good way to make testing truncated SVD compatible with finite-difference checks, since this more or less requires that the variations are small enough that the truncation selects the same subspace every time.
I think the only way around this is to carefully craft input matrices with well-defined spectra, which can and should be done in a separate PR