Need M combinatorial for existence of injective model structure on $M^G$?

It seems very unlikely to me that you will be able to get any useful handle on the generating acyclic cofibrations in the injective model structure, even in simple cases like when $I$ is the delooping of a group. The only way I have ever seen to show that they exist is by using some nasty cardinality argument akin to Lurie's A3.3.3.

I believe that the first construction of the injective model structure on diagrams of simplicial sets (specifically) was in Alex Heller's monograph "Homotopy Theories", section II.4. I don't quite understand his argument at the moment; it doesn't seem to use cofibrant generation directly.

Another, somewhat more general, reference, which is also earlier than Lurie, is Tibor Beke's paper Sheafifiable homotopy model categories, which uses a logical approach and requires that the model category be not only combinatorial but "sheafifiable".

I don't think I've ever seen any construction of an injective model structure for a non-combinatorial model category.

Edit: Apparently Lurie's construction has been studied further in an abstract context by Makkai and Rosický and Makkai and Rosický and Vokřínek.

It seems this question has been answered very nicely since I asked it, in the paper Left Induced Model Structures and Diagram Categories (Contemp. Math. 641 (2015) 49-81). They prove in Proposition 4.17 and Theorem 4.19 that if $M$ has a Postnikov presentation and the class of cofibrations coincides with the class of monomorphisms, and if $D$ is a small indexing category, and if maps in $M^D$ can be factored into a trivial cofibration followed by a fibration via the Postnikov presentation on $M$ then $M^D$ admits an injective model structure. In particular, this does not require $M$ to be combinatorial. You can also drop the need for cofibrations to equal monomorphisms if you ask $M^D$ to have both types of factorization. This paper also does a fantastic job spelling out the duality between asking a model category to be cofibrantly generated vs. to have a Postnikov presentation. Since the injective model structure is dual to the projective, I don't think a better answer than this one can be found, but I am glad to know combinatoriality is not needed. For those who enjoy the Bayeh et. al. paper, I also recommend the extension of this paper which can be found in the appendix to Hess and Shipley's Waldhausen K-Theory of Spaces via Comodules.

EDIT: A follow-up paper, A necessary and sufficient condition for induced model structures (Journal of Topology, 2017) gives an answer completely in terms of factorization systems, and again avoids the need for $M$ to be combinatorial. This is the best I could have hoped for when I asked the question way back in 2012.

This is probably not the sort of answer that the original question was going for, since you said you were happy to assume cofibrant generation but not combinatoriality (i.e. not local presentability), but it may be useful to other readers. Namely, if you're willing to keep local presentability and instead relax cofibrant generation to accessibility, then there is now a very general result about the existence of injective model structures in the paper Injective and Projective Model Structures on Enriched Diagram Categories (arXiv:1710.11388) by Lyne Moser.