Q_TERM is empty after adding required environment variables

### Checks

- [x] I have searched [github.com/aws/amazon-q-developer-cli/issues](https://github.com/aws/amazon-q-developer-cli/issues?q=) and there are no duplicates of my issue
- [x] I have run `q doctor` in the affected terminal session
- [x] I have run `q restart` and replicated the issue again

### Operating system

Linux 6.8.0-1032-aws - Ubuntu 22.04.5 LTS

### Expected behaviour

Attempting to do initial setup of amazon q cli.

### Actual behaviour

When running `q doctor` it initially requested adding some environment variables to the sshconfig, i did that and attempted to run `q doctor` again to confirm the solution and am met with
`✘ Amazon Q terminal integrations: qterm is not running in this terminal, please try restarting your terminal

Q_TERM=

✘ Doctor found errors. Please fix them and try again.
`
With no indication of how to resolve the issue.

### Steps to reproduce

_No response_

### Environment

```yaml

[q-details]
version = "1.12.7"
hash = "ed77799a52644ba49a3ed76b15c25cd9334c4b8d"
date = "2025-07-22T17:39:11.331361Z (2d ago)"
variant = "minimal"

[system-info]
chip = "Intel(R) Xeon(R) Platinum 8259CL CPU @ 2.50GHz"
total-cores = 1
memory = "7.64 GB"

[system-info.os.linux]
kernel_version = "6.8.0-1032-aws"
id = "ubuntu"
name = "Ubuntu"
pretty_name = "Ubuntu 22.04.5 LTS"
version_id = "22.04"
version = "22.04.5 LTS (Jammy Jellyfish)"

[environment]
cwd = "/home/USER"
cli-path = "/home/USER"
os = "Linux"
shell-path = "/usr/bin/bash"
shell-version = "5.1.16"
install-method = "unknown"
in-ssh = true

[env-vars]
PATH = "/var/efs/opt/nvm/versions/node/v18.19.0/bin:/usr/local/sbin:/usr/local/bin:/usr/sbin:/usr/bin:/sbin:/bin:/usr/games:/usr/local/games:/snap/bin:/home/USER/.local/bin:/home/USER/release/release/portal/util:/home/USER/release/release/portal/dev/site/bin:/home/USER/release/release/portal/audit:/home/USER/.local/bin:/home/USER/.local/lib/node_modules/.bin"
Q_SET_PARENT_CHECK = "1"
SHELL = "/bin/bash"
TERM = "xterm-256color"
XDG_SESSION_TYPE = "tty"
```