Model category structure on spectra
Since you tagged this as a reference request, let me give you some relevant references. The first injective model structure for motivic spectra that I am aware of is Jardine's paper Motivic Symmetric Spectra. It's written in the context of simplicial sets (i.e. motivic spaces start as simplicial presheaves). A more general approach was given by Hovey in Spectra and symmetric spectra in general model categories. It works for any left proper, cofibrantly generated model category (cellular or combinatorial), with an endofunctor to stabilize (in this case, smashing with the motivic sphere).
To get more fibrancy, you need your underlying category to have all objects fibrant. So, you should use topological spaces instead of simplicial sets, and take the projective model structure on presheaves (so all objects are fibrant). Instead of dealing with set theoretic issues around topological presheaves rather than simplicial presheaves, it's best to start with the category of $\Delta$-generated spaces instead of all of $Top$. It's a combinatorial model category with all objects fibrant: see the notes of Dan Dugger. Where he assumes Vopenka's principle, you can instead follow this awesome answer of Tim Campion.
You can use the machine of Jardine or Hovey with $\Delta$-generated spaces as the starting point, but there's a problem. First, to even get motivic spaces, you need to do a left Bousfield localization, after which the fibrant objects will be the Nisnevich local objects. Secondly, to get to the stable injective model structure, you need another left Bousfield localization (classically, the fibrant objects are the $\Omega$-spectra).
Thinking classically, the model structure for spectra that has all objects fibrant is $S$-modules, from EKMM, specifically VII.4.6. A motivic version is given by Po Hu. She defines a model structure on $\mathcal{U}$-spectra, where $\mathcal{U}$ is a universe (for you, it would be the infinite dimensional affine space over your base). In Section 7, she proves that her category $Spectra(\mathcal{U})$ is Quillen equivalent to Jardine's, and in Sections 11-13, that her model categories $\mathbb{L}$-spectra and $S$-modules (of motivic spaces) are also Quillen equivalent to the same. So the homotopy category of each of these models is what you want. The paper is written simplicially, but you could re-do it using $\Delta$-generated spaces instead of simplicial sets. The category of $\mathbb{L}$-spectra is a category of operad-algebras in $\mathcal{U}$-spectra, so the fibrations are underlying fibrations.
Bringing this back to the context of your question, if you start with $\Delta$-generated spaces and use stabilization machinery or $S$-module machinery, you are almost there, but I still don't know how to deal with the Nisnevich localization (see Theorem 5.5 of Hu's paper).
Is there a model structure on Spt(S), having SH(S) as homotopy category, such that every object is fibrant? If so, could you provide a reference?
No, if the given model category of spectra Spt(S) (for which there are many different, but Quillen equivalent, definitions) is not right proper, because if all objects are fibrant, the model category is necessarily right proper.
Yes, if we are allowed to pick a specific Spt(S). According to Corollary 2.21 in a paper by Nikolaus (Algebraic models for higher categories), every cofibrantly generated model category C where all trivial cofibrations are monic is Quillen equivalent to a model category where all objects are fibrant.
Thus, one can take the projective or injective model structure on simplicial presheaves, localize it, pass to spectra (which gives one possible choice for Spt(S)), and then apply the cited result to get a desired model category.