Agreed - power supply caps aren't gonna fix a bad gain/impedance structure. The cost reduced design of the Fostex 3180 has probably prevented them from using a low noise amplifier to handle the pickup coil of the reverb tank, and that IMHO is the most fruitful way to reduce the total device noise. Driving the reverb tank is not gonna be that tough using a simple IC op amp, but the pickup coil is going to be a low impedance, low level source, not much different than a ribbon mike. Typical low cost IC op amps will have an equivalent noise impedance of around 1 or 2kΩ, and if that coil is closer to 50-100Ω, then there's a lot of noise from the electronics that does not have to be there.
Not sure which reverb tank they used, and what the source impedance of that pickup coil is, but chances are that a newer, low voltage noise op amp such as the LME49710 might work well, or perhaps the even lower voltage noise AD797. Another idea is to use a low noise common emitter discrete stage wrapped inside of an op amp feedback loop to reduce noise even further.
While this sounds slightly complex, it actually might not be that difficult. Given that one is replacing an 8 pin DIP package, a new carrier board could be made containing a low noise discrete front end and an SO8 package op amp, and it'd likely be easy to get it to fit into something close to an 8 pin DIP footprint. Bias current induced dissipation might be a problem with a very low voltage noise discrete stage, but a little 'glue-on' heatsink might make this work well enough.
Of course, there may be some silly design problems at the driving end of the device, but my vote is to provide the slightly expensive low noise pickup amplifier stage that Fostex chose not to provide, in order to grind the noise down, to the limits of the reverb tank that Foxtex chose.