I think you're talking about Cardfight Vanguard since I believe that there's already an anime series named Battle Spirits. It premiered last week and I could see 4Kids being interested in picking up the license, but I could easily see it going to another company as well.
Sonic X and DBZ Kai do seem to do well more so than the other shows on the block, so I could see 4Kids trying to get a couple more action-based series. I can understand why they've kept in that toyetic series formula given how most of their most successful shows, Pokemon, Yu-Gi-Oh! DM, Sonic X and arguably Dinosaur King, as it was really popular outside of the US, were toyetic and they seem to do really well with merchandised based series. Given that interview from a few months ago where Mark said they didn't really know what they were getting into with One Piece, I wouldn't be surprised if they were kind of nervous, but I agree that an action anime with little editing involved for them would be a good choice. I think they could have a better chance going for another DBZ Kai situation where they simply get the broadcasting and streaming rights to the series and then have to do little to no edits with the series.
Even if the new CEO wanted to have 4Kids make their own shows again, they'd still need more money to make that happen. I wouldn't mind seeing 4Kids create their own shows again since most of them were really fun, but I don't think that will happen anytime soon. For now, I think that it's better for the company to acquire licenses to series since I believe that is overall less expensive than creating an entire series from scratch.
Well that is an interesting idea. While an animated series that follows the storyline to the Inheritance Cycle more so than the movie did sounds kind of cool, along with seeing the magic, dragons and other character in animation form, I don't know how likely that could happen. Not only did the movie bomb, but I'm not sure if the book series is even that popular to warrant an animated series. If Harry Potter couldn't get one in its hay day, I don't think that the chances of Eragon getting one are that good. They'd have to reduce some of the fighting and death in the later books, as well as the romance between Roran and Katrinna. It has interesting potential, but I don't see it as a likely scenario to happen.
I thought that Eragon, the book, was much better than the movie adaptation. It wasn't groundbreaking material, but it was fun and enjoyable fantasy read. While Eldest did drag on a bit here and there, Christopher Paolini did a better job than I expected with focusing on the point of view of two characters and it was still overall good. With Brisingr though, it dragged on way too much in the middle for me to care that much about the little bits of events relevant to the plot stuck in the beginning and ending of the book. It's a decent book series at best, but I don't think it would be something kids could look forward to watching on Saturday morning. I think that there are a lot of other better developed fantasy series out there that would appeal to kids more.
As for the permission from FOX, the movie was distributed by 20th Century Fox, so they might hold some short of license over the name Eragon. If that was the case, then 4Kids would need to get their permission to do an animated series around the book series. I still doubt that they would go that route with this series in particular, but I'm guessing that's what they'd have to do with FOX.