No fuse will save a FET or any modern power stage component, and if it needs to do that in the first place, the thing is broken by design. Fuses wear out if you operate it close to the rated value, it will nuisance-open eventually. Rate the fuse appropriately, treat is as a safety device only, and you wouldn't need to make it replaceable.
Slight clarification, there are some components that can be fuse protected effectively; but only one is likely to be found in a typical small VFD (the input rectifier), and none of the active components (MOSFETs or IGBTs). And weakest links being what they are, it's the latter ones that matter.
Some interesting technologies such as the UR fuses noted, but these are outside the range of what i'm running (0.75kw) and overkill it would seem. Although I would be interested in reading more about this technology as it's promoted as being very high speed, able to "protect power semiconductors and DC circuits", "good resistance to cyclic loading" and "No downgrading of fuse characteristics over time". How this is achieved and what it means in reality is another story - there seems to be little information out there.
May find more info under the general term, "semiconductor fuse" -- so called because they act fast enough to manage to protect some types. Namely, semiconductors with low voltage drops and high surge ratings: basically just diodes, SCRs and TRIACs. Things used for general AC power control, rectification to DC (at the input of most electronics), and industrial power converters. (These devices can be used to switch many of the same loads that transistors are used on, but they act slower, so are typically only used as a last resort, at very high voltages and power levels where transistors are too bulky/expensive -- consider a VFD delivering 4.8kV, and about as many amperes, to a motor of ungodly horsepower.

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The fuses are typically constructed something like, a parallel array of silver foils, punched in a diamond checkerboard pattern, and filled around with pure silica sand. The diamond cutouts causes current to neck down to narrow spots, which are able to heat up quickly; meanwhile the remaining diamond patches give wide area between necks, able to dissipate nominal heat into the sand filling. When melting and subsequent arcing occurs, the silica melts and vaporizes, the vapor quenching the arc while the bulk absorbs blast energy. Typical clearing times are as low as single to fractional milliseconds -- impressively fast for any kind of mechanical device. It's also much shorter than a mains cycle, meaning they can break the circuit at any time, without waiting for the AC cycle to cross zero -- which means they are often suitable for DC power as well (and usually rated as such). (Most fuses continue arcing during a full peak, until mains voltage falls to zero then reverses, the low point giving a chance for the arc to cool and extinguish; high voltage DC arcs are dangerously persistent, so this is a bit of a built-in safety feature of our AC system.)
Note that, when faulting occurs, massive currents are drawn. For residential circuits, up to a few kA (breakers are typically rated to clear 10kA); for industrial circuits, up to 100kA, or large circuits even more (e.g. at substations or generators). This heats the fuse rapidly, causing it to clear quickly.
Fuses don't act quickly at all, in general. Your average fuse will handle about twice rated load for some seconds before melting; it might be guaranteed between a minimum of a few seconds, and a maximum of several minutes! Fuses are not precision devices, and cannot be relied on for that kind of limiting or protection. They are, indeed, there to protect the wiring and such -- the wires in the walls will heat up much slower than the fuse, so aren't in any danger of starting a fire (hopefully, anyway!) by the time the fuse opens.
Anyway, fault current. So, keep in mind, as current goes up, fuses open faster, and these semiconductor fuses can open in a ms or so, but that's at typical fault currents (~kA). However, transistors (MOSFETs, IGBTs, etc.) cannot switch into such currents: they are current limited devices, they don't handle faults nearly as boldly as their diode-ish kin. What'll happen is, instead of dropping a couple volts as in normal operation, the transistor sees full fault voltage -- DC supply or mains peak, say 320V or more -- at whatever maximum current it is able to draw, say 10s or 100s of A. It heats up very quickly, and in as little as 10-20µs for IGBTs, or maybe 100µs for generously sized MOSFETs, the silicon die surface begins to melt, at which point it can't turn off no matter how hard the control tries, and the game is over.
Semiconductor fuses are sometimes still used with such devices -- in this case, they're not being used to protect the device, but to reduce the amount of carnage created. When a transistor dies in this manner, the die surface first melts, then the bondwires (inside the package) connecting it to the power source melt and vaporize, and within a few hundred microseconds, a hot, dense ball of plasma is building inside the package. Within a millisecond or so, the package goes off like a bullet, fragments of which may be propelled at dangerous speed. So, guards for shrapnel and arc flash are a good idea in the design of such equipment. (This is why we keep the covers on before throwing the switch...

) Well, with a semi fuse halting that by the ~1ms mark, shrapnel may be avoided, as well as major arc flash damage to the circuit; repair may be possible, but mind that, during that miniature fireball, high voltage and current were applied to all terminals of the transistor -- usually destroying the drive circuitry too. (So you'd need to replace at least the power board, which will be most of the unit anyway, so... a unit like yours is probably disposable. Larger industrial kit can be worth making with more replaceable pieces, though.)
Whereas with a conventional fuse, or breaker even: after the failed device explodes, the arc flash will continue between terminals (and anything else remotely conductive nearby, including circuit traces, component terminals, screws and nuts, etc.), until power is cut -- some 10s of ms later. In which time, huge swaths of circuitry can be charred and eroded, and the whole thing is a loss; let alone if the covers were open and someone happened to be standing beside it(!).
Meanwhile, the advantage of transistors is their fast reaction time -- a well-designed control circuit can detect such a malfunction, and disable operation within microseconds. Maybe one transistor still fails, say due to overheating; but the other transistors around it can actually be saved, and maybe just a few components need to be replaced (transistor and driver?).
The downside about complex electronics, of course; there's no way to tell what kinds of things they put into the design. It works in the average case, fine, but what happens under extreme conditions? And if those conditions should occur due to errors in the system itself (e.g. software control), how could those conditions be triggered and tested? Well, you can't know, in general. And even if you know about these things (like many of us here do), it's a huge amount of work to dig into the thing and figure out what it's doing. So it's just the same old crap shoot for anything you get these days -- if it works, it works; if it blows up, replace it (with a different brand, perhaps).
(Just to be clear, arc flash isn't much of a hazard on residential circuits. Typically you'll get a flash and a pop, and that's it. Never hurts to wear PPE, of course. It's definitely a good safety lesson if you'll be working in industry; no need to be fearful, just respect it and follow protocol. Cheers!)
Tim