Were to get fuse box for dedicated line?

The reason you're having a problem finding a load center type fuse box is because (I'm pretty sure) they're not made anymore - they're sort of like NOS, surplus only.

Where are you going to grab the power for your fuse box? If it's from the main house panel, then the feeder wire from the main panel to the fuse box has to be protected; the only way to do that is to install a new breaker for this feeder. You can't run an unprotected wire outside the panel, if you do, your fuse box becomes meaningless. If you have to use this breaker, then why not just make it a circuit instead - less devices along the power path. Either way, I would check with the local building department to cover yourself.

... I was assuming a remote panel. Installing a parallel, or subfeed panel, is a legitimate installation provided that the subfeed panel is properly bonded, grounded and supplied with neutral lugs for each circuit. In other words, it has to be a UL listed load center type panel - not a switched fuse, gem box or a metal vanilla box with fuse holders. Further, the wiring to the panel has to be taken off the main lugs downstream of the main panel breaker and run through a bonded conduit fitting only.

Another snag is that if you do find a load center type fuse box, you must use the edison s-type base fuse (tough to find in ceramic). Bussmann still makes the standard W edison bases in ceramic; but these are not allowed in new load center installations - the "penny argument" - just in switched fuse boxes for furnaces, etc.

Some installations do have a circuit breaker provided by the utility in the meter pan but most of the time the meter is protected by what are called limiter lugs. These are, basically, inline coupling devices with a fusible link, meant to protect the utility wires only.

As far as breakers v fuses - breakers have more copper in the power path than plug fuse; albeit a twisted, convoluted pathway. Maybe it's breaker contact pressure that's responsible for degrading the sound, I don't know. But, fuses definitely protect better than breakers by limiting current inrush much faster.