Symmetrical power supply: Rail fuses useful or dangerous?
Fuses are not intended to protect the circuit, but to prevent a possible fire. While the 135W that one rail could deliver regularly is allready sufficient to start a fire, the energy in the case of a short curcuit would be much higher.

Your question leaves open whether it is a single copy or a prototype for a whole series.

Development
Building an amplifier and then turning the power switch on is a very bad idea. As a teennager a friend gave me a 1000W amplifier kit and instructed me to assemble it. I did so - when I turned the switch around, there was a loud bang and light went off. The mains fuse, output transistors, driver transistors and two multimeters - all toast.


Either you need an adjustable dual power supply or a variac.
Then you must short the audio input (no signal) and put ammeters on both power rails.
Now you can slowly increase the voltage starting from zero. Watch continiously the ammeters and control the circuit for hot parts and smoke signals.
Switch power off, wait until the capacitors are empty
Connect the speaker and redo point 3.
Switch power off, wait until the capacitors are empty
Connect audio source with volume 0
Switch power on
Raise slowly the audio volume while whatching the circuit.


Finished amplifier circuit
Now it's time to analyze the circuit and look what happens when one half of the power goes away. I would pay particular attention to whether the speaker could be damaged.

Now one must consider that if the fuse triggers, something went broken anyway and a repair is inevitable. For a series product you could decide that two fuses (plus one for the primary side of the transformer) are enough and everything else is too expensive. If you're a thrifty guy you could even decide that a single fuse on the primary side is sufficient. For your individual item you can of course take further protective measures.

I don't see any other feasable option than the two already named in the comments to your question.


Two mechanicaly coupled circuit breakers
A voltage monitoring circuit that triggers a thyristor to blow the other fuse to