CI - Test that quickstart-chat builds

[bfops]

Description of Changes

Test quickstart-chat in CI. This is for #182.

It turns out we had mostly-working code to do this, but it was commented out.

API

• This is an API breaking change to the SDK

No breaking changes

Requires SpacetimeDB PRs

None

Testing

• The new CI passes on this PR
• If I introduce a type error in quickstart-chat, the CI fails (https://github.com/clockworklabs/spacetimedb-typescript-sdk/actions/runs/16155839095/job/45597982694)
• If I edit the generated bindings in quickstart-chat, the CI fails (https://github.com/clockworklabs/spacetimedb-typescript-sdk/actions/runs/16155815395/job/45597907808)

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Latest commit: 514405a

Merging this PR will not cause a version bump for any packages. If these changes should not result in a new version, you're good to go. If these changes should result in a version bump, you need to add a changeset.

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commit: 514405a