I was wondering if there is a mechanism that can give priority to
certain genes to be accurately duplicated. Some sort of trigger that
says "double-check this specific gene before continuing with the
A mechanism that would say: "double check this part, it's important" isn't known to me neither, but if I understood your question correctly I think the answer to your question lies somewhere else.
Let's start with the mutation rate you mentioned, some sequences are harder to replicate by DNA polymerase (it makes more mistakes there, such as in highly repetitive sequences, methylated cytosines etc.), so the mutation rate is higher in these segments (they are called mutation hot spots), but otherwise the DNA polymerase makes errors randomly, throughout the whole genome. These are in better case repaired. No special care given, repair polymerase tries to repair everything.
Now, to my point. What if the repair fails? In genes, that are critical to cell survival, development of organisms etc. (for it's life) you really see less mutations (if any) present. (and I believe this was the reason for the question, correct me if I am wrong.)
Moreover, these places in genome can be highly conserved between different organisms. More here. What this says is that nature doesn't allow cells/evolution/whoever to experiment (or make errors) on these positions. If mutation occurs here and isn't repaired, the cell simply dies and therefore you cannot see the mutation. The reason for death may be accumulation of toxic protein, which cell is not able to get rid of with disrupted enzyme, or lack of energy, because it cannot oxidize fuel... there are plenty possible lethal outcomes. For some genes, the cell can substitute their product by another protein, but some are essential.
(In multicellular organism can substitute for it, unicellular won't care because it's dead already)
I believe this is the mechanism, by which cell can distinguish between important and "less important" genes you were asking for.