The Protoflorian initially halves all elemental damage, but about a minute into the battle, it will start "Exoproofing" itself, shifting its elemental affinities to absorb one element, but making itself weak against the opposite.
A good strategy is to have Lightning use Libra, possibly twice, at the start of the battle. The default paradigm should be set to Supersoldier (COM/SYN) so that Hope can buff Lightning and himself with Protect and Shell. Afterward, the party can shift to Slash & Burn (COM/RAV) to start building the chain gauge. When the gauge reaches around 120%, the party should switch to Dualcasting (RAV/RAV) to build it up faster.
If Aster Protoflorian uses either Exoproofing Thunder or Exoproofing Water, the player should take advantage of it by having Lightning switch to Ravager and use Water/Aquastrike or Thunder/Sparkstrike respectively. The War & Peace (COM/MED) paradigm can be set up so that Hope can heal. Once the creature is Staggered, the party can start doing some real damage (Launch from Lightning's Commando role helps), though it will likely not be defeated during the first stagger.
If Aster Protoflorian uses either Exoproofing Fire or Exoproofing Ice, Hope can switch to Synergist to buff with Barfire/Barfrost to reduce damage taken.
SummoningOdin while Aster Protoflorian is staggered is an option, but is usually not needed because constant launched attack chains will do more damage overall.