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 this paper by Lyne Moser.

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.