[Bug]: Upgrade from production build to RC build does not update build number in Manage Extensions

### Describe the bug

When upgrading from 13.22.0 production build from MetaMask website to RC build 13.25.0, the build number in Manage extensions does not update, still displaying 13.22.0

### Expected behavior

Build number to update in Manage extensions

### Screenshots/Recordings

https://github.com/user-attachments/assets/c3df4c77-bdc4-4bca-85f7-1ca62602666c

### Steps to reproduce

1. Download MetaMask from [website](https://metamask.io/download)
2. Complete onboarding
3. Download RC build
4. Drag and drop RC build in Manage Extensions
5. Notice build number in Manage extensions is still 13.22.0

### Error messages or log output

```shell

```

### Where was this bug found?

Internal release testing

### Version

13.25.0

### Build type

None

### Browser

Chrome

### Operating system

MacOS

### Hardware wallet

_No response_

### Additional context

_No response_

### Severity

_No response_

[Jira issue](https://consensyssoftware.atlassian.net/browse/MCTP-132)

### Expected behavior

No response

### Screenshots/Recordings

No response

### Steps to reproduce

1. Download MetaMask from [website](https://metamask.io/download)
2. Complete onboarding
3. Download RC build
4. Drag and drop RC build in Manage Extensions
5. Notice build number in Manage extensions is still 13.22.0

### Error messages or log output

```shell

```

### Where was this bug found?

No response

### Version

13.25.0

### Build number

No response

### Build type

None

### Browser

Chrome

### Operating system

MacOS

### Hardware wallet

No response

### Additional context

No response

### Severity

Sev2

Reproduce the upgrade in Chrome by installing the production build, then dragging the RC build into Manage Extensions. Trace where the displayed extension version or build number is sourced, and verify that Manage Extensions shows 13.25.0 after the upgrade instead of 13.22.0.

Written by the indexing model from the issue text.