Handle transient database-not-found during managed Dolt restart

## Problem

When a managed Dolt server restarts, a client can briefly observe the server as reachable while a database probe reports "database not found". In gastownhall/beads#4449, the proposed beads-side workaround retries that DB-not-found probe with exponential backoff before falling through to bootstrap/init.

That restart tolerance is real and useful, but the retry belongs at the Dolt driver/storage boundary rather than in beads bootstrap code. Beads should not need storage-engine-specific sleep/retry loops to distinguish "database is genuinely absent" from "the server is still coming back after restart".

## Why this should live in the driver

Beads can only infer this state indirectly from driver/server errors. If beads owns the retry loop, every caller has to duplicate driver-specific knowledge and tune backoff around managed-Dolt restart behavior.

There is also a concrete product cost to doing it in beads: in shared-server mode, a first-ever bootstrap for a new project on a machine whose shared server already hosts other project databases can eat the full backoff window before falling through to init. The submitted beads-side loop was 10s + 20s + 40s.

## Requested behavior

Please consider one of these driver-level fixes:

- make transient restart/not-ready states distinguishable from a genuinely absent database;
- retry known transient DB-not-found probes inside the driver with bounded backoff; or
- expose an explicit driver option/capability so higher layers can request restart-tolerant open/probe behavior without hardcoding Dolt-server error handling.

## Source context

This came from the Layer 2 portion of gastownhall/beads#4449. Beads plans to keep the Layer 1 URL guard from that PR, but route the managed-Dolt restart tolerance here rather than landing a beads-side storage workaround.

_codex-gpt-5.5-high on behalf of matt wilkie_

No files, tests, or driver entry points are named. Start at the driver/storage boundary described in the issue and read the restart context from gastownhall/beads#4449. Done means transient managed-Dolt restart states are handled or exposed by the driver, without a beads-side storage-specific retry loop.

Written by the indexing model from the issue text.