The crooks at immigration desk will stop you from boarding the plane if you do not produce Shram Karyalaya letter. There are three workarounds:

If you are short on time - I mean very short on time, buy a one way ticket to Delhi for earlier that day, get boarding pass for your US flight and Delhi flight, using separate desks and at separate times the same day. For all documentation purposes, show your Delhi boarding pass. But once you cross immigration, use the US destined boarding pass. Kinda lengthy process. but worked for me recently.

Option 2: Keep a a few (about 5-10 Rs 1000) notes handy, and keep a few of those in your passport if the immigration guy raises noise about Shram Karyalay letter.

Option 3: Fly overseas from Delhi instead. Problem solved. BTW, this is why a lot of women (and men too) trafficcers use Delhi or Mumbai airports as their take off points.

But if you have time, you are better off going to Shram Karyalaya (who will also send you to get a Life Insurance, etc.). Spend a few hundred/thousand rupees on the Lekhandaas and the 'fast-forwarders' who will get your work done in no time.

Good luck