I think the Hero ship should have more interaction with the primary aspects of the main game, rather than being relegated to only getting upgrades and new stuff from the nebulae. In general, the harder the game is the more opportunities the hero should have to find upgrades outside the nebulae. Below are ideas for how to integrate Zenith, Neinzul, and Spire upgrades as rewards for doing stuff outside the nebulae.
Another of the problems with Hero ships is that the costs of unlocking are problematic. If you get to level 10, you can have one fully upgraded Large module and one fully upgraded Small module, which gives your ship little in the way of variety, and does not really encourage experimentation or multi-weapon deployment. The suggestions below seek to address this as well, though in the random, unplanned way that characterizes technology unlocks in the normal game.
If you want to balance the Hero ship with Fallen Spire and the regular game, have all retrieval missions for Fallen Spire provide global XP to the hero and have the number of spire facilities control the unlock of certain ships and modules. Retrieval of the refuge ship would give all heroes access to Spire frigates AND the lowest mark of a random spire modules, and then whenever a new spire ship class is available to build for the normal player, the Shadow (Spire) version of that ship class becomes unlocked for the hero, along with the lowest mark of a random spire module. This is purposefully more powerful than the Zenith and Neinzul options below, because the power level necessary in a Fallen Spire campaign over its entire course is significantly higher.
If you have Golems (Hard) turned on, the normal player capturing a Golem should provide all hero players with (randomly) either a Zenith shadow vessel of the lowest available unlocked mark (so no unlocking a Zenith Destroyer if you don't already have another destroyer hull unlocked) OR the next mark of a random Zenith weapons module. For Golems (Medium) it could instead have a random chance of providing upgrades (50% or lower?)
If Hybrid Hives are turned on, have it so that each level of Hybrid Hive is worth a certain level of "salvage". When enough salvage is collected by the hero, it unlocks (randomly) the lowest applicable mark of Neinzul hull (same restrictions as above in the Zenith example) or a random Neinzul weapons system. Unlike the previous two suggestions which are not necessarily based on the hero's actions, this suggestion prompts the players to aggressively pursue the Hybrid Hives, which I think is fitting because of the Hybrid Hives using a different logic that allows them to function more intelligently than the typical AI forces.
Edit: Human/generic upgrades could be gained by rescuing rebel human colonies and destroying marauders. Destroying Golems and Spirecraft in exo waves should also contribute to unlocking Zenith and Spire modules, I suppose.
With all of the above turned on, heroes won't actually have to do nebulae missions to advance, which I think is an important choice to be able to have. There will still be incentive to do so (free hulls and new weapons modules, plus knowledge for any normal players). Without Fallen Spire they'll need to do nebula missions to get the next tier of ships, but it will still provide flexibility to hero players by giving them a broad range of mission choices and make hero play a little less one dimensional.
One other thing I would really like to see is a way for Normal players to assist Hero players in nebulae in some minor way so that it doesn't feel like we're playing two separate games, with me just watching and waiting for the nebula mission to clear up so I can move on with hero assistance. Perhaps include an unlockable ship, the Shadow Transport (shipcap 1 or 2 to prevent easy shipping of your entire fleet at once, 2500 knowledge cost), which can carry a small number (25? 20?) of ships and traverse the nebula wormhole? Say that the reverse-engineered shadow drive is so large it could only fit on a transport, and with significantly decreased transport capacity at that. That way I (as a normal player) can join a Hero player in one of these battles, but ships I send will generally be hopelessly outclassed on their own.