[Feature Request] Support API-based authentication for Chrome Extension

Bug Description
I have API access to Claude and the Chrome Extension is not yet avaible. I would love to have it working with the API too.

Environment Info

• Platform: win32
• Terminal: windows-terminal
• Version: 2.1.42
• Feedback ID: 813185bd-fba7-46aa-9226-31b06fde6029

Errors

[{"error":"Error: ENOENT: no such file or directory, open 'C:\\Users\\conse\\.claude\\remote-settings.json'\n    at openSync (unknown)\n    at <anonymous> (B:/~BUN/root/claude.exe:12:1109)\n    at BP (B:/~BUN/root/claude.exe:11:49471)\n    at readSync (B:/~BUN/root/claude.exe:12:1037)\n    at FP (B:/~BUN/root/claude.exe:6173:113)\n    at TX (B:/~BUN/root/claude.exe:6175:226)\n    at _IA (B:/~BUN/root/claude.exe:183:138169)\n    at sE$ (B:/~BUN/root/claude.exe:183:138332)\n    at Az0 (B:/~BUN/root/claude.exe:184:1202)\n    at bLH (B:/~BUN/root/claude.exe:184:2080)","timestamp":"2026-02-14T01:47:26.700Z"},{"error":"Error: ENOENT: no such file or directory, open 'C:\\Users\\conse\\.claude\\remote-settings.json'\n    at openSync (unknown)\n    at <anonymous> (B:/~BUN/root/claude.exe:12:1109)\n    at BP (B:/~BUN/root/claude.exe:11:49471)\n    at readSync (B:/~BUN/root/claude.exe:12:1037)\n    at FP (B:/~BUN/root/claude.exe:6173:113)\n    at TX (B:/~BUN/root/claude.exe:6175:226)\n    at _IA (B:/~BUN/root/claude.exe:183:138169)\n    at sE$ (B:/~BUN/root/claude.exe:183:138332)\n    at FD (B:/~BUN/root/claude.exe:183:140069)\n    at nn (B:/~BUN/root/claude.exe:184:2290)","timestamp":"2026-02-14T01:47:26.744Z"},{"error":"Error: ENOENT: no such file or directory, open 'C:\\Users\\conse\\.claude\\remote-settings.json'\n    at openSync (unknown)\n    at <anonymous> (B:/~BUN/root/claude.exe:12:1109)\n    at BP (B:/~BUN/root/claude.exe:11:49471)\n    at readSync (B:/~BUN/root/claude.exe:12:1037)\n    at FP (B:/~BUN/root/claude.exe:6173:113)\n    at TX (B:/~BUN/root/claude.exe:6175:226)\n    at _IA (B:/~BUN/root/claude.exe:183:138169)\n    at sE$ (B:/~BUN/root/claude.exe:183:138332)\n    at FD (B:/~BUN/root/claude.exe:183:140069)\n    at nn (B:/~BUN/root/claude.exe:184:2290)","timestamp":"2026-02-14T01:47:26.748Z"},{"error":"Error: ENOENT: no such file or directory, open 'C:\\Users\\conse\\.claude\\remote-settings.json'\n    at openSync (unknown)\n    at <anonymous> (B:/~BUN/root/claude.exe:12:1109)\n    at BP (B:/~BUN/root/claude.exe:11:49471)\n    at readSync (B:/~BUN/root/claude.exe:12:1037)\n    at FP (B:/~BUN/root/claude.exe:6173:113)\n    at TX (B:/~BUN/root/claude.exe:6175:226)\n    at _IA (B:/~BUN/root/claude.exe:183:138169)\n    at yZA (B:/~BUN/root/claude.exe:2131:20664)\n    at d8I (B:/~BUN/root/claude.exe:2131:21569)\n    at <anonymous> (B:/~BUN/root/claude.exe:7460:510)","timestamp":"2026-02-14T01:47:26.750Z"},{"error":"Error: ENOENT: no such file or directory, open 'C:\\Users\\conse\\.claude\\remote-settings.json'\n    at openSync (unknown)\n    at <anonymous> (B:/~BUN/root/claude.exe:12:1109)\n    at BP (B:/~BUN/root/claude.exe:11:49471)\n    at readSync (B:/~BUN/root/claude.exe:12:1037)\n    at FP (B:/~BUN/root/claude.exe:6173:113)\n    at TX (B:/~BUN/root/claude.exe:6175:226)\n    at _IA (B:/~BUN/root/claude.exe:183:138169)\n    at sE$ (B:/~BUN/root/claude.exe:183:138332)\n    at FD (B:/~BUN/root/claude.exe:183:140069)\n    at yzH (B:/~BUN/root/claude.exe:324:1160)","timestamp":"2026-02-14T01:47:26.753Z"},{"error":"Error: ENOENT: no such file or directory, open 'C:\\Users\\conse\\.claude\\remote-settings.json'\n    at openSync (unknown)\n    at <anonymous> (B:/~BUN/root/claude.exe:12:1109)\n    at BP (B:/~BUN/root/claude.exe:11:49471)\n    at readSync (B:/~BUN/root/claude.exe:12:1037)\n    at FP (B:/~BUN/root/claude.exe:6173:113)\n    at TX (B:/~BUN/root/claude.exe:6175:226)\n    at _IA (B:/~BUN/root/claude.exe:183:138169)\n    at sE$ (B:/~BUN/root/claude.exe:183:138332)\n    at FD (B:/~BUN/root/claude.exe:183:140069)\n    at uzH (B:/~BUN/root/claude.exe:324:1765)","timestamp":"2026-02-14T01:47:26.756Z"},{"error":"Error: ENOENT: no such file or directory, open 'C:\\Users\\conse\\.claude\\remote-settings.json'\n    at openSync (unknown)\n    at <anonymous> (B:/~BUN/root/claude.exe:12:1109)\n    at BP (B:/~BUN/root/claude.exe:11:49471)\n    at readSync (B:/~BUN/root/claude.exe:12:1037)\n    at FP (B:/~BUN/root/claude.exe:6173:113)\n    at TX (B:/~BUN/root/claude.exe:6175:226)\n    at _IA (B:/~BUN/root/claude.exe:183:138169)\n    at sE$ (B:/~BUN/root/claude.exe:183:138332)\n    at FD (B:/~BUN/root/claude.exe:183:140069)\n    at zPA (B:/~BUN/root/claude.exe:919:34299)\n    at async <anon…

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