This sounds fascinating. I can see two potential pitfalls, but I suspect they're navigable.
The first issue is that you might have to mess with the results tables and turf systems if players are going to be members of different factions. The actual mechanics of Blades tend to assume that you're getting help a decent percentage of the time, which means that the fact that you don't have a good chance of success isn't so bad. Similarly, every character on the team is working together during downtime, which makes recovery something that different people can sacrifice an action to at different times without a sizeable problem. If everyone has to manage their own group's faction, you'll want to make some fairly substantial changes there.
The second question - what does a 0-card pull actually look like? Since there's advantage to not always taking the best card, you can't take the Blades approach of "draw two, take the worse," but presumably you want there to be a chance of success. Is it a system where tempting fate uses the Major Arcana, seperated out into a different deck, or is there something simpler for it?