This thread prompted me to have another look at the Hyperion and, well, that ended with a complete redesign of the ship. While it went in a different direction than proposed here, the thoughts and ideas in this discussion were very helpful, so: thank you all!
Patch notes for the new Hyperion:
Hyperion: redesigned almost entirely; new sprite
Reclassed as "Heavy Frigate"
Weapon slots: 2 medium energy and 1 medium universal
All weapon slots are forward-facing hardpoints
Flux stats increased to almost high-tech light cruiser levels
Armor and hull values increased significantly
Speed and maneuverability decreased to be on the low end of the frigate range
Phase Teleporter:
Requires zero-flux bonus to be active to be used
Ship is vulnerable during wind-up
Removed flux cost
Added 3 second cooldown
AI adjusted to be more convervative with its use
(Old AI still available for use by mods)
The idea is that the teleporter becomes useful for two things: moving the ship around quickly (more player impact on battles!), and finding a good spot to get "stuck in" and unload its firepower (a decision where skill is involved in making a good one).
Importantly, it's not very useful for getting
out of trouble, so if you get the "when and where to commit" decision wrong - especially combined with the ship's low-ish speed - you're likely to be heavily punished. Basically, thought about exactly how to engage over quick reflexes.
The new ship *is* quite tanky, with the 0.6 shield and good flux stats, but given that it's usually punching at something bigger, that doesn't count for quite as much as it might seem. In a way, it's almost like an extreme phase ship - move quickly, engage anywhere - but without the ability to get out of dodge easily. And not really feeling like a phaseship to play.
PPT-wise, the base is still 120, but you can increase that by something like 4 minutes when all the options for doing that - including some new skills - are combined.
Perhaps obviously, with its 3 hardpoints, it's also very weak to fighters - and even just being swarmed by faster frigates - so it doesn't seem like a Hyperion mono-fleet would be desirable, even if one is mainly sticking with frigates.
(Some further tweaks still possible, of course, but the above is feeling pretty good to fly!)