If verification are not a part of your services, you don't do verification. However, what your describing is a claims denial or a rejection (clearinghouse level) and if claims follow-up and denials is a part of your service, it should be your responsibility to collect the correct information, even if not given to you at the time of initial submission. What you are describing really isn't a part of verification and eligibility but rather a follow-up/denial. It would also depend on your contract. My contract lists every single specific service I include for full practice management. I ALWAYS include verification/eligibility because then I know it's done right and how I want it which in turn keeps those rejections down.
To avoid this and any other services you might want to include but feel your not getting paid for, and you are doing a flat fee, why not base your flat fee on an hourly rate? For example if the client has you working an average of 30 hours a week on their account and you want $25 an hour, their flat fee would be $750 and then add expenses. A sliding scale to accommodate new patients (added work) could be based on an average of demographics/claims you can enter in an hour. If you can do 6 in an hour (demographics and claims) on an average, your sliding scale would be an additional $25 for every six new patients. Very easy. This is how I do it.