A couple of things:
1) Try to avoid loading BAM files directly into the browser in order to check expression levels. BAM files are not normalized at all, and even if they have the exact same read counts this is still not representative. If you want to check browser tracks then normalize them properly as suggested here A: ATAC-seq sample normalization (quantil normalization). The post talks about ATAC-seq but the same holds true for RNA-seq, just consider "regions" as genes.
2) It is not surprising that lowly-expressed genes are not significant at n=3. You will probably need many more replicates to have the power to call them DE. Can you show the output of plotMA and indicate where your genes are that you are interested in?
3) Doing without FDR correction is probably not okay. Yes, that would not be acceptable. Do more replicates if you are interested in lowly-expressed genes. The rule is simple, large effect sizes and genes with high expression require fewer replicates to have the same power as lowly-expressed genes and/or lower effect sizes.
4) I was thinking about using another software... Please not. Proper statistics needs replicates, and DESeq2 is absolutely fine. Your study is probably simply underpowered.
n=3 is in fact a bare minimum for a DE analysis so you cannot expect to even get close to the power to find all DEs for lowly-expressed genes.
Is there anything, I can do, other than telling the wetlab to re-sequence deeper?
Don't sequence deeper, do more replicates this is far more important than read depth. You can explore the read depth effect by simply multiplying your raw counts by factor 2, 3, 10...and see how things change. This is of course a bit artificial but I doubt that you gain anything at n=3 by just sequencing deeper. You need more replicates.