@notallwrong sorry, I was on a short vacation... I'll make the promised update to my answer by this Friday at the latest.

@notallwrong I'll try as well to translate some instances of the consistency theorems Willie asked for above into the language of cutoff functions. I don't mind losing the bounty due to time. Regarding your points (1) and (2), I believe that what goes wrong in your example is that typically consistency theorems imply some convergence conditions for the matrix elements that seem at first glance to be violated in your case. I'll try to clarify that as well within the next few days...

@WillieWong I'll try to do that in the next few days, as time allows...

The nontransverse intersection example is very interesting, but it raises quite a few questions on its own regarding the concept of "negative manifold dimension". For instance, when does the relation to topological and Hausdorff dimensions break down, and how does that happen?

In due time: if you want to get the spin-statistics connection I've mentioned at the end of my answer, you do need full Lorentz/Poincaré covariance because otherwise you cannot define the spin of the field, which is linked to the representation theory of the Lorentz group. Another Wightman axiom I didn't include in my answer out of irrelevance to the OP is that of uniqueness of the relativistic vacuum state, encoded in a cluster property for the truncated $n$-point functions (a sort of decay towards infinity in spacelike directions) which is automatic for generalized free fields.

There are many books on the subject, but the go-to reference for this material is R.F. Streater, A.S. Wightman, PCT, Spin and Statistics, and All That (Princeton University Press, 2000). I've omitted the axiom of Lorentz/Poincaré covariance because it plays no role in the context of the OP - in fact, only translation covariance (which is a consequence of the spectral condition) is relevant here.