Yeah my buddy was told the samething from New Britain. I would bet this is an artificial time line. But i dont know how you can fight it. Try going to the station and ask them when they do fingerprinting (my town only does it on monday morning). Tell them you have two reasons, one being a job. I was fingerprinted before I worked for my neighbors daycare when I was younger, and bus drivers are also required to be fingerprinted. Make sure you have all the paperwork filled out when you go (and notarized). Bring your money order And bring the letters of reference, some stations don't require them, but since they are already giving you problems don't make it easier for them to refuse you.
This way when you go get fingerprinted for your "job" you can have them take your paperwork at the same time. For your job make sure you tell them you need to take the set with you.
You other option is if you have relatives in a different town, fill out your paperwork as if you live their. Go to that towns PD and have them send the papers to your relitives address. Then when you go to get the permit you file it as if you have moved. This does work, worst case, is you must call the dmv and tel them to change your drivers license adress (no need to get a new ID) Then just change it back. This works because some people have multiple residents with in our state.