When I have to get a document (or anything small really) from foreign, I only use DHL or FedEx. I generally don't use the overnight option. This document sounds critical, so having someone else bring it down for you would be cheaper, but then you would have to reply on them mailing it to the Ministry. And then the Ministry would have to send it back to them, between the name difference and time to to get it back, they may already be gone. There are way too many points where a simple mistake would ruin it.

Unfortunately this is one of the things that you should not be worried about the money as much as making positive it happens. DHL is a reliable courier service and can handle all the signatures and such, so they are probably the option I would use.

Three years ago I had to get some docs from the US, and it was US$80. It was worth it for the peace of mind.