I think the way to asnwer the feeder -vs- branch circuit issue is:
Would the wires be adequately protected if a non fused disconnect was used downstream? If so, its a branch circuit.
The reason a fuse is added would be because the item said fuses and not an HACR breaker. Article 440 allows any hermetic air conditioner can have overcurrent larger than the wire (up to the code limited percentage over) because the thermal cutout in the compressor helps to protect the wires.
In the case of a hot tub it is similar, you may have the correct size circuit, but it is not GFCI. A GFCI doesn't protect wires, but protects people. Putting a GFCI breaker based disconnect in line with this circuit shouldn't change it to a feeder. This is important because pool and spa feeders requires special wiring methods (insulated grounds, and conduit) whereas branch circuits aren't as restrictive.
The rules allowing an HVAC branch circuit to have a breaker larger than the wire ampacity only applies to branch circuits and not feeders. So again, it is important to determine if this is a feeder or a branch circuit in HVAC circuits. It seems kind of silly that this is the case, but it becomes less silly if you realize that putting a breaker on the end with different characteristics doesn't change things much.
I must admit though, that I don't know if a circuit using a circuit breaker is adequately protected if the HVAC appliance says fuses only. Is that fused disconnect added at the end protecting the wires, or the HVAC unit? If the breaker was the normal size for the installed wire, I'd say of course. But HVAC circuits typically are breaker over the normal 310.16 ampacity, so now I'm not sure (depends on the HVAC thermal cutout characteristics). If you installed an HVAC unit that allowed breakers and you used a disconnect with a breaker, then this is a branch circuit.