It'd be a lot more than 60C - the goal is to keep the material from cooling past the melting point by the time it's been deposited, and thus the important factor is the rate of energy loss, which is dramatically accelerated in a temperature differential of, say, 650C instead of say 145C - so I'd guess you'd want about 150C - 300C difference.
I'd bet inconel and other high temperature alloys would be eroded very quickly, anything that's fluxed enough to melt below 1000C is going to be extremely corrosive. Hot molten sodium hydroxide levels of corrosive. Fun to think about though, a serious materials challenge for sure.
I'd guess that it's a lot easier to maintain the whole build chamber at 500° than to maintain the hotend at 850°, but I haven't tried it.
Felsic lavas (and magmas) which melt at those temperatures do not typically contain a lot of alkali oxides, but they do contain some. See https://en.wikipedia.org/wiki/Calc-alkaline_magma_series#/me... However, ferrous and quasi-ferrous alloys like inconel are among the best choices for alkali corrosion. For example, table 4 in Birgitte Stofferson's dissertation https://orbit.dtu.dk/en/publications/containment-of-molten-n... gives an inconel corrosion rate of 1.06 mm per year in molten NaOH at 600°, which happens through oxidation from oxygen dissolved in the melt. Monel 500 corroded only 5.06 mm per year at 700°.
If you were trying to keep a 100μm hotend aperture within a ±10% tolerance, you could start with a 95μm aperture and replace the hotend when the aperture had expanded to 110μm. At 1mm/year those 15μm would be 5 days of printing time, which seems like a usable hotend lifetime. Presumably printing in lava rather than 100% NaOH would extend the lifetime further.
I think that if most of the things in the vacuum chamber are at room temperature, while the lava filament is at 700°, that won't substantially reduce the radiative heat loss. If almost everything inside the vacuum chamber that isn't mirror-coated is at something like 500° or 600°, I think it would work. Maybe that could save you from having to keep the walls themselves at 500° or 600°.
I'd bet inconel and other high temperature alloys would be eroded very quickly, anything that's fluxed enough to melt below 1000C is going to be extremely corrosive. Hot molten sodium hydroxide levels of corrosive. Fun to think about though, a serious materials challenge for sure.