How is this the case?  Mining generates new Bitcoin.  Printing FRNs (or creating FRN denominated loans electronically) creates new fiat.  How is that 'the opposite'?

I fear you are ascribing causality here where there is none.  I have seen nothing to suggest mining (i.e creating new currency) 'increases the value of each bitcoin'.  Indirectly of course the process of mining; the fact there is a mining network is what gives bitcoin value therefore it is fair to say without the process of mining the existing bitcoin would be of no value but that is not the same.  The usd value of bitcoin is determined by the balance of supply and demand.  Nothing else.  If it's going up it's because demand is outstripping supply.  Increasing coin production by more mining (if that were not restricted by the process) would increase supply and usd value would decrease.

As I said above Bitcoin is currently out-inflating USD and most other currencies and the only reason that isn't putting people off Bitcoin is that we can know with full certainty it will eventually be a deflationary currency.  It is far from one for the few years ahead.  If the FRB were to find a way of encoding a promise to do the same (halve the introduction of new currency every x years) into the way they create new dollar Bitcoin would lose a substantial part of its advantage.  However, apart from the logistical implications of that I can't see them or the US (or any other) government ever agreeing to yield the monetary control they currently wield.