From what we've been able to research forth, there is no such thing as a stripped-down asgard core. There is having an asgard core, and there is not having one. No partials.
Removing one bit form it, will cripple other bits. For a warship designed to go into hot situations where it might be at risk, there might be reason to skip having a full core, but it could just as well have a powerful self-destruct built into the memory banks to keep information out of wrong hands.
Tau'ri may know more than a little about Asgard tech, but there is still a lot that is far beyond them. Much like the database of the ancients was unimaginably far beyond the Asgard.
Till we see lore-based proof that they've managed to build an asgard core that wasn't flawed yet had removed the knowledge of the Asgard, we'll treat the ship core as the same as the one on the Hammond and on the Odyssey. Probably with more intricate security systems to prevent abuse, but even these will probably have work-arounds.
And if you're thinking of suddenly removing the core, Sep.... That is beyond illogical on a research vessel. We may joke about using the core for all sorts of DEM tasks (Deus Ex Machina), but we'd not actually do it. Using DEM is boring and shows that a real solution wasn't bothered with. Pretty sure most others here feel the same.
By this time, the ships will be built with the core inside them from the start. Its less time-consuming than post-construction retrofit, and it'll make a better ship in the end, as you won't need to remove/replace half the systems to get it in right. Its possible that even the Apollo and the Daedalus have cores by now, but that isn't specified anywhere we could find, so we'll go with that those don't. Maybe to keep the risk of Asgard knowledge falling into wraith hands at minimum. They would really not want them to gain any science even remotely related to the intergalactic hyperdrive tech, which would certainly be in the core.
Its one thing to have the knowledge available. Quite another to be able to navigate it easily or even use it without deadly mishaps.
There's two things we feel the ship should not have:
ZPM: There's few of these, and the dedicated warships & such need them more than a research vessel.
Cloak: Even with years to study the Odyssey cloak that Daniel-Merlin added, they probably don't understand enough about it to duplicate it.
As for balancing out the enemies like the Lucians:
This is simple. The goa'uld hardly ever innovated much from Ptah's original ship specs. When they did, they kept their improvements so secret that most would die with their creator. Only those with knowledge beyond that of the goa'uld (I.E. Anubis) were able to implement major stuff, and quite a bit of that was lost with him.
Humans on the other hand, be they Tau'ri or not, tend to be quite innovative. And there's a number of sources for tech out in the galaxy that could balance the scales somewhat. It doesn't matter much what the source is, so long as it gives a trace if a balance.
When it comes to ensuring that people don't go for the lazy stuff like metagaming, trust the players to police each other, then slap the ones that don't get in line.
