chore(mcp): convert lib/local-branch.js to TypeScript (Phase 2 of #7291)

[JSONbored]

Problem

#7291 proposed migrating packages/loopover-mcp to TypeScript, phased by file size. This is Phase 2:
lib/local-branch.js (592 lines) -- substantial enough to be its own batch, smaller and lower-risk
than bin/loopover-mcp.js (Phase 3).

Proposal

Convert lib/local-branch.js from .js to .ts, building on Phase 1's (#7328) build pipeline.
No existing .d.ts to remove -- this package has none.

Deliverables

• lib/local-branch.js converted from .js to .ts
• CLI still runs identically post-build

Test Coverage Requirements

No behavior change from this migration alone. Existing tests must continue to pass unmodified.

Expected Outcome

local-branch.js is real TypeScript, clearing the way for #7291's Phase 3 (bin/loopover-mcp.js).

Links & Resources

• chore(mcp): migrate packages/loopover-mcp from plain .js to real TypeScript #7291 -- the parent migration proposal and phasing plan
• chore(mcp): add tsc build pipeline + convert small lib utilities to TypeScript (Phase 1 of #7291) #7328 -- Phase 1, the prerequisite build pipeline this depends on

[linear-code]

JSO-5115