If the amp's cap terminals are merely another power input connector (ie, same as +12V & gnd supply), then no problems because the fuses are downstream (between in&battery/cap and the amp).
But if it was for internal filtering of an internal 12V supply....
But that should still have downstream fusing... (should!)....
And it doesn't make sense anyhow - internal voltages are usually far higher than 12V, and if they are 12V, why not have the cap "with the input supply"? (Though I can see some logic - but that really needs 2 sets of fuses etc etc etc.)
As for the HC600, AFAIK it is an AGM/VRLA battery.
But it is probably a rebadged whatever, or may be cheap crap - I haven't seen any detail.
I am surprised it claims low ESR yet does NOT supply a figure.
And "600W". For how long? What is its AH rating?
I wouldn't buy one unless I knew more...
It matched AGM 12V-18AH dimensions (18AH @ 20-Hr rate) and might be a UB12180 (AUD$35) or Yuasa NP18-12 (AUD$50) - both with changes terminals.
As for using a low ESR battery in place of a cap, welcome to a huge group that reckon that is the sensible thing to do - unless you like bling or $pending big.
I have oft maintained that a $20 12V-1.2AH else a more "economical" $40 12V-7AH quality AGM will be far superior to a big cap (probably 10F and beyond - last calcs I saw suggested 30F or 40F for a 7AH).
With similar ESRs (internal aka (Equivalent) Series Resistance), they are in simple terms identical in respect of "low ESR" responce.
Then we compare what happens to the cap once it dips to battery voltage (say 12.7V) compared to the battery - that's when the cap effectively no longer exists (you can often start a car on a 7AH battery, and certainly an 18AH AGM. Will a 100F cap crank & start a car?)
And above the battery voltage - ie, say 12.7V to the system/alternator voltage), I maintain that that the battery's "surface charge" exceeds the energy/power of caps (10F, 100F etc; but depending battery size).
Most other arguments are probably wrong, else they are only DC related (not steady-state or AC response - ie the long note or 2nd or later cycles of the "dipping" note etc), and I have yet to see (meaningful) supportive data such as oscilloscope pics or true-RMS with peak figures etc.
But you have a great and correct idea. (IMO)
BTW - if the cable's fuse blew, whether cap or battery, the amp still has power.
But the cable's fuse (at the start of that cable way upstream somewhere) is to protect the cable, not the load.
The amps fuses are to protect the amp.
An internal "short" or overload may not be enough to blow the cable's fuse, hence equipment protects itself (if justified - eg amps, TVs, PCs).
The main consideration is how you connect your two batteries together - because one bad or mismatched battery can drag the other down and that can destroy both batteries.
I prefer a relay that only connects the batteries together when charging. At other times, the batts are separate so your amp can flatten its bat, but you can still crank & start, and then charge the 2nd batt.
BUT there must be a fuse at EACH END of that inter-cabling - ie, each battery has a fuse (usually from its + terminal) as close as practicable to the battery (but maybe not above the battery...??) in case any downstream cable part faults (shorts to ground etc).