What is my recourse if an ER doctor failed to diagnose 1 of 2 work comp injuries? on first visit?

It depends on two things:
1) Whether the misdiagnosis was unreasonable, or, to put it a different way, whether any reasonable, properly trained medical care provider would have been expected to come to the correct diagnosis. The law accepts that medical care is an art, not a science--sometimes good doctors who do everything right still get it wrong. The error must have been unreasonable, something no doctor would be expected to potentially make, to make the care provider liable.
2) Whether the delayed or wrong diagnosis hurt you in any way--e.g. caused your injury to get worse. Even when a doctor or other medical care provider makes a mistake, they are only liable for the costs or injuries caused by the mistake. If you had more or less the same outcome with the mistake as you would have without it, you cannot get any compensation. 
Bear in mind that medical malpractice cases can be expensive: you need to hire a doctor or doctors to test/examine you, write a report, and testify if needed, and in case that goes all the way to trial without settling, that can be thousands of dollars. You'd also realistically need an attorney--it is very unlikely that a nonlawyer could successfully bring this kind of case. Unless you suffered serious adverse effects from the wrong diagnosis, it is not likely worth taking legal action.

Medical malpractice is negligence.  Negligence is the failure to exercise due care (that degree of care that a reasonable medical practitioner in the community would have exercised under the same or similar circumstances to prevent foreseeable harm). Negligence on the part of the hospital is the failure to exercise due care (that degree of care that a reasonable hospital would have exercised under the same or similar circumstances to prevent foreseeable harm).
Prior to filing a lawsuit for negligence against the ER doctor and the hospital, it may be possible to settle the case with their malpractice insurance carriers.  
When you complete your medical treatment and are released by the doctor or are declared by the doctor to be permanent and stationary, which means having reached a point in your medical treatment where no further improvement is anticipated, obtain your medical bills, medical reports and documentation of wage loss.  Your claim filed with the malpractice insurance carriers for the doctor and hospital should include those items.
Compensation for the medical bills is straight reimbursement.  The medical reports document your injury and are used to determine compensation for pain and suffering which is an amount in addition to the medical bills.  Compensation for wage loss is straight reimbursement.
If the case is settled with the malpractice insurance carriers for both the doctor and hospital, NO lawsuit is filed.
If you are dissatisfied with settlement offers from the insurance carriers, reject the settlement offers and file your lawsuit for negligence.
If the case settled with one, but not both parties (doctor and hospital), only name the party with whom the case has not settled as a defendant in your lawsuit.
If the case is NOT settled, your lawsuit for negligence must be filed prior to the expiration of the applicable statute of limitations or you will lose your rights forever in the matter.