These are two different requirements, not the same thing. And to me, both are easy to answer according to your description
First is to prove that you have work authorization, so the two H1 I-797/I-94 (from both companies) will suffice. Or if you have c9 EAD, you are also fine for this requirement.
Second is employment verification letter to prove that the LC same or similar job offer is still good. If you are working for a company that is sponsoring you H1, I don't see why they would refuse to write such simple letter? (unless it's NOT a similar job so they don't want to lie.)