This is very related to some work in progress of mine, which began as part of a larger project of Aaron Mazel-Gee and Markus Spitzweck. Details can be found in the long version of my research statement, section 5.1. If you want more details, please email me separately, as there are still things I want to work out before making a draft public.

I am much more accustomed to model categories so let me phrase my answer in that context. Basically, one way to get a model structure for stable motivic homotopy theory is to use Mark Hovey's stabilization machine from Spectra and Symmetric Spectra in General Model Categories. This gives you a model structure on motivic symmetric spectra which generalizes the usual one on symmetric spectra. Classically, in order to get the strictification you desire (which operad people might call rectification), you would need to pass to the positive stable model structure on symmetric spectra, introduced in Brooke Shipley's A Convenient Model Category for Commutative Ring Spectra. The basic point of this model structure is to force the unit to no longer be cofibrant, removing Gaunce Lewis's obstacle to having a good model category spectra. In this model category, commutative monoids do inherit a model structure (with weak equivalences and fibrations maps which are such as maps of symmetric spectra). Furthermore, a cofibrant commutative monoid must be cofibrant as a symmetric spectrum (this is what "convenient" means). Finally, rectification holds as shown for example in Theorem 1.4 in Elmendorf-Mandell Rings, Modules, and Algebras in Infinite Loop Space Theory.

The theorem in my research statement shows that Hovey's machine can be tweaked to output a positive stable model structure, at least when the input model category is combinatorial. In particular, we checked that this applies to motivic symmetric spectra. To apply the Elmendorf-Mandell result it seems you must also assume the model category is simplicial, i.e. satisfies the SM7 axiom. Thankfully, motivic symmetric spectra does satisfy SM7. Determining in what generality these rectification results hold is still future work for me, so I don't want to say anything too definitively on this yet. For sure both $E_\infty$ ring spectra and strict commutative ring spectra inherit model structures, and it seems highly likely they are Quillen equivalent, so you have a good hope for rectification coming from these considerations.

EDIT: It has been pointed out to me that this answer was slightly lacking in a couple of places. For one thing, the idea of the positive model structure originally goes back to Jeff Smith, so I should have included his name above. Secondly, the real feature of importance in the positive model structure is that for any cofibrant spectrum $X$, the natural map $(E\Sigma_n)_+ \wedge_{\Sigma_n} X^{\wedge n} \to X^{\wedge n}/\Sigma_n$ is a weak equivalence, i.e. the map from the the homotopy colimit (which is the extended powers) to the colimit (symmetric powers). I believe I can prove this for motivic symmetric spectra, so the rest of Shipley's paper should go through (I'm still writing this up, though, so for now you should treat this as hearsay). If you're interested in learning more about this property, I asked a couple of questions about it on MO and got some very informative answers from Peter May.

I should also mention that classically if you instead work with orthogonal spectra rather than symmetric spectra then you also need to pass to a positive variant in order to get the property above, and hence rectification. See Mandell-May-Schwede-Shipley Model categories of diagram spectra. This has also been done for equivariant orthogonal spectra in a paper of Mandell-May. I don't know anything about orthogonal motivic spectra and I don't know how that would be done, because motivic spaces are built on simplicial sets rather than topological spaces, and I don't know how to represent $O(n)$ in that setting. Classically, if you use S-modules then rectification comes for free because commutative spectra and $E_\infty$ spectra are the same thing (see EKMM, the point is that the monoidal product already builds in higher homotopies). Po Hu has developed motivic S-modules but does not seem to mention commutative monoids or $E_\infty$ at all in her paper. I don't know that side of the story well enough to know if it still comes for free or not.

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