The choice mostly has to do with how many false leads that you want to risk tracking down in your follow-up studies and how much you want to investigate the age-associated differences in the gene-protein associations.
In this type of study it's not completely clear to me (as a biologist rather than a statistician) how well the assumptions underlying the Benjamini-Hochberg procedure to control false discovery rate (FDR) hold. Biologically, at least, the changes among levels of 3000 proteins as a function of your gene of interest are unlikely to be independent. I tend to think of FDR values as being a guide to future study rather than a definitive statement of which differences are "significant."
There's also no magic in a 5% FDR (or, for that matter, in the 5% family-wise error rate used so widely as a measure of "statistical significance"). If you are willing to risk a bit of extra work in follow-up studies, a 10% or 20% FDR could be defensible and could lead to keeping more true positives.
For publication, so long as you are clear about how you proceeded your choice shouldn't matter. If it does matter to a reviewer or editor, then you can always revise.
It sounds like you are planning to do separate analyses for each age group. I'd recommend instead evaluating the interaction between the gene and age in a single analysis. That will more directly tell you which proteins are differentially changed depending on age. If you have actual ages, you might consider moving away from binning by age and instead model age flexibly and continuously with splines. In the age-grouped data that single analysis will end up with 9000 tests, but you can always choose a different FDR to compensate with respect to designing follow-up studies.
