So far, it does not seem so.
As for why, her projections seem to be different from Cat's. There's no reason all projections have to work the same.
Personally, I think it's because Cat's are a very real replica of herself, whereas Isabelle's are a replica of anyone. Even if Isabella chooses to project herself, it seems like it might be "less real" and not carry her metagen.
Edit: A while back, I said this regarding Isabella, Cat, and testing how that worked:
I was only 80% joking. A sample size of 1-2 means practically nothing.
If we really wanted to test it thoroughly we'd want dozens of different replicators/protectors at a minimum. Until then, any conclusions could be considered highly speculative and preliminary at best.