The details of the chemistry control need to be worked out, but the reprocessing in a LFTR is very straightforward. The most common and problematic fission products are gaseous and just bubble out of the liquid fuel, which by the way is 750C, not several thousand degrees C. Some plate out, and others may even remain dissolved in the salt for the lifetime of the reactor.
If you are concerned about human error, then there is no safer place for radionuclides than fluoride salts. Poor chemistry control may damage the reactor, but the fuel and non-volatile fission products are going nowhere, even if the reactor vessel is cracked open. At worst, some of the fission products in the off gas system could escape, but they pose no long term health concern--the ones that do remain safely locked away in the salt, which will eventually freeze solid.
The reactor you describe is a fantasy, and wasteful besides. Ironically, unless it were a molten salt reactor, it could never burn through 100% of the fuel, producing a long-term radiological hazard that should not be left in the ground. An MSR burns virtually 100% of the fuel, is walk away safe, and requires no complex or safety critical control systems. It essentially runs itself, following the load; you add fuel and remove heat, that is all. Sure, some maintenance is required, and a lot of engineering, but nothing else holds half so much promise.