New channel: Desktop @ picker lags after invite; Mobile has no invite for people/agents

## Bug Description

On a self-hosted community with a remotely hosted agent (not managed on the mentioner's Desktop install):

1. **Desktop:** creating a channel and inviting the agent now works, but the composer `@` picker does not list that agent immediately after the invite. After a short wait, the same picker starts showing the agent and mentions work.
2. **Mobile:** channel create works, but there is still no UI to invite a person or an agent into that channel. `@` mentions still work.

These look like two membership-surface gaps around the same job: add someone to a new channel, then address them.

## Steps to Reproduce

### Desktop — picker lag after invite

1. Create a new channel on Desktop.
2. Invite a remotely hosted agent into that channel (this step itself now succeeds; it previously did not).
3. Open the composer in that channel and type `@`.

### Mobile — no invite

1. Create a new channel on Mobile.
2. Look for a control to invite a person or an agent.

## Expected Behavior

- Desktop: as soon as the invite succeeds, the invited agent (and invited people) appear in that channel's `@` picker. No extra wait / restart should be required.
- Mobile: after creating a channel, the same invite-people / invite-agent action Desktop has should exist.

## Actual Behavior

- Desktop: invite succeeds, but `@` does not list the agent at first. After some time (no other action reported), the agent appears in the picker and can be mentioned.
- Mobile: channel create succeeds. There is no invite for people or agents. Mentions still work.

## Environment

- Date: 2026-08-20
- Surfaces: Buzz Desktop (Nest) and Buzz Mobile
- Topology: self-hosted community; agent runtime on a different machine from the mentioner's Desktop
- Local reference host (not necessarily the mentioner's Desktop): Buzz.app `0.5.17`

Exact mentioner Desktop / Mobile build numbers were not captured in this report. Happy to add them in a comment.

## Additional Context

Adjacent, not a duplicate of the older “picker hides cross-owned agents” cluster:

- https://github.com/block/buzz/issues/3125
- https://github.com/block/buzz/issues/2508
- https://github.com/block/buzz/issues/3277
- https://github.com/block/buzz/issues/3204

Those were “agent never appears in Desktop `@`.” This report is: **invite to a new channel is now possible on Desktop**, then membership/picker is stale for a while; **Mobile still cannot invite at all**, even though mentions work.

Please omit asking for community URL / channel UUID in public comments; repro is the membership + picker surfaces above.