(Point of this email: A strict Deligne spectrum gives a canonical hofib. Such a Delige spectrum might be obtained via E-infty methods on the Beilinson product.)
I chatted a bit with Denis-Charles. Briefly, he told me at Regensburg a way to obtain a canonical homotopy fiber of H_Beilinson —> H_Deligne, on the level of the model category (as opposed to the homotopy category). The whole question relies on the following idea of his: find a complex that calculates Deligne coho and is a commutative associative unital algebra (on the nose). Call this complex D. There is an equivalence
Ho(CommAlg)_Beilinson = Ho(H_Beilinson-CommAlg)
Notation: CommAlg is the category of (strictly) comm, ass. algebras in the model category underlying SH. Ho(…) is its homotopy category, and Ho(CommAlg)_Beilinson is the subcategory of those objects acted on by H_Beilinson (the motivic coho spectrum). Right hand side: H_Beilinson is a chosen representative (on the model category level; this can be done thanks to a proposition in their paper), so it makes sense to talk about commutative H_Beilinson-algebras, and one can take its homotopy category (this is the right hand side). Imagine for a second we have such an algebra D as above. I.e., it is an element in CommAlg. By what we have done so far (etale descent), it is also an element in Ho(CommAlg)_Beilinson. Hence, we get an object in the right hand side.
There is a Quillen functor
H_Beilinson-CommAlg —> H_Beilinson-Mod (\phi : H_Beil –> X) \mapsto hofib (\phi)
It passes to the homotopy cat’s
Ho(H_Beilinson-CommAlg) —> Ho(H_Beilinson-Mod) = DM_Beilinson
Starting with our (\phi_Deligne: H_Beilinson –> D) in the left hand side, we get an object \widehat{H_Beilinson}, in a canonical way. The only choice we made in this construction was the choice of representative of H_Beilinson, but this choice does not matter, the same way as the equivalence (of which the above one is a sibling)
Ho(CommAlg)_Beilinson = Ho(H_Beilinson-CommAlg)
does not depend on the choice of H_Beilinson-representative (the left hand side does not see the choices, the right one a priori does).
This is the starting point. (The mentioned equivalence does not seem to be in their long paper, but if I understood him right, it follows by similar methods.) It is worth pointing out that the commutativity of the algebra D is not used in this claim, so this would also work for (strictly) associative unital algebras. Recall that Beilinson (see Esnault Viehweg) gives products \cup_\alpha on D such that \cup_{0.5} is commutative, \cup_{0} is associative and has a left (or right, I don’t remember) unity, while \cup_{\alpha = 1} is also associative on the nose and has the unitality on the other side. Which means, very close, but not quite as good as we need. Secondly, Burgos provides a complex with a product that is strictly unital and commutative, but only associative up to homotopy. So, none of the two work for us.
So, the question is: how to get a representative of Deligne cohomology which is a commutative associative unital algebra (all three on the nose). In the 2nd email (see 2nd attachment) Denis-Charles explains a way to obtain such an algebra structure. I just received it two hours ago, but it looks like the key point is to use the model category of commutative (ass.) algebras in the model category of sheaves (and then spectra, I suppose). According to D-C, the algebra structure on Betti cohomology is only possible with rational coefficients; it is possible though to go down to Z-coefficients by replacing algebras in all of the above by E_\infty-algebras.
I have to say, I feel quite embarassed that he just comes up with these things as a little exercise. (Not least, he also seems to give another proof that the classical Beilinson regulator and the one we get using their theorem agree.)
nLab page on Jakob email March 2011