This would break the immutability of soulbound badges/tokens. One of the benefits of immutability here is that once assigned, applications can aggressively cache results because they have high confidence that a token cannot be removed, reassigned, moved, mutated, changed, etc.
If they are not part of this standard, then they should be moved to a different EIP. I did consider this, but I felt that it was important to emphasize the immutability constraints within this EIP so it was clear that these badges were immutable through and through. It is worth noting that the content of the URI is delegated to future EIPs, but I’m hesitant to move even more.
Re: name/symbol: I feel like this information should live at the URI. It is descriptive metadata about the token, and these tokens are likely useless without the metadata (e.g., without image/description/details). If readers are going to have to lookup some metadata externally anyway, it feels like all of the token metadata including name/symbol should live there rather than being split on and off chain.