The purpose of normalization is to unify the dimensions. The key is range analysis, independent of the method.

The edgeR group says that you can use RSEM expected counts for their DE workflow: https://support.bioconductor.org/p/65890/

DESeq2 works by internally normalizing by library size. This is built into the model, so I really don't see a way to use it properly with normalized counts. There may be a workaround if the data is already normalized by the effective library size, but I haven't seen anything out there to help with that in the 5 years I've been using the software. http://bioconductor.org/packages/devel/bioc/vignettes/DESeq2/inst/doc/DESeq2.html#why-un-normalized-counts

The voom transformation tool found in the limma package followed by a limma differential expression test could in theory be used if you set lib.size for each sample to 1, again provided that the normalization was based on library size... though I would always be cautious about trying to use packages based on modeling and then altering the model right from the get go. You are more likely to force your data into a poorly fitted model and get false readouts than you are to actually accomplish your goal.

How has the data been normalized? I know you said it isn't trivial, but if you can find the way to do it, it might be easier in the long run if you write a quick script that allows you to reverse the normalization of all your samples before piping it into a DESeq or edgeR workflow.

The simples thing would be to just run a t-test, LIMMA, lm() or a mixed model on the normalised data. If the sample size is big enough it will not make much of a difference whatever you choose and how you tweak the models.

If you want to get back to raw values you need to know which normalisation was applied. I guess that is almost impossible because different researchers e.g. apply different filtrations before calculating library sizes etc. Some scale the library sizes and others do not before calculating RPKM/FPKM. Some use voom transformations etc. Not easy at all.

I would either ask for the raw data, or just go with the normalised data using more standard tools.

If you can't get the raw data to process and get your conts, it's very hard to transform normalized values back to raw counts.

But it is quite straight forward to use the normalized data and building your model. It normally gives you more freedom - if you are confident enough - to adapt it and have a better fit to the data. You can tweak and play around with your co-variables and so on.