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The UK Home Office has no provision for issuing a travel document under such circumstances. That only occurs or applies to those who cannot use or get a passport from their home country, e.g., refugees, asylum seekers.
While you can request that your documents be returned to you, that may invalidate your request for further leave to remain.
As an Indian national, you could apply for an emergency passport through your own country. However, such circumstances may not qualify for issuance, particularly as it may cancel your current passport (and which has your US visa).
As inconvenient as it is, you may have to postpone or cancel your travel outside of the UK.