As far as IRS is concerned, the money that came to you as a reimbursement is just that and is not taxable to you. The company is just wrong to claim it as income to you (yes, they are being greedy and ignorant). This is where it is handy to be claiming ALL of every payment and then deducting ALL reimbursements and unreimbursed expenses.
Lets say I have $10,000 in total receipts and $1000 is from the company with the flaky 1099 and $1000 is a 1099 from a company that did it right. I list the 2 1099s for a total of $2000, just as shown on the 1099s. Now I put in all other receipts as a single item, so there is the other $8000.
I happily go on down the form listing equipment, supplies, mileage, yadda yadda, and when I get to Part V of the form, "Other Expenses", that is where I deduct all reimbursed business expenses as a single line item titled "Reimbursed business expenses claimed as income" and on the line below pull out all of the flat fee and other unreimbursed business expense as "Unreimbursed business expenses".
You do not submit copies of the 1099s with your return, you are appropriately claiming EVERYTHING and then subtracting out what you are entitled to subtract out. The numbers all eventually work and you should be golden.