To answer the question you asked - yes, you may bill only a portion of the total service provided to the insurance. However, there are more important factors.
If you are participating with the insurance plan, it is extremely likely that your contract would prohibit you from billing a patient for a covered service simply because the reimbursement is insufficient. Your practice can chose not to provide a particular service, but you generally can't charge the patient if you don't like your contracted fee schedule.
If it is a non-covered service, you may charge the patient (and should provide an estimate). Whether or not your practice submits the coding for a non-covered service is up to your own internal policy and/or patient preference.
If you are not participating with the insurance plan, then you have no contract with the payor and may charge the patient for any and all services.
And as
@Robyn07 mentioned, you want to be sure that coding both services is appropriate based on the scenario and documentation.
Let's use a dermatology example. Patient comes for a rash on their arm. Provider evaluates and treats the rash. While patient is there, she notices your office also provides cosmetic botox and provider injects botox in her forehead as well. You may bill the E&M service to insurance, and have patient pay up front for the botox service which is not billed to insurance. If that same patient came in 3 month later for another botox without an additional problem, then it would not be appropriate to bill an E&M service to the insurance.
If that patient came in for a rash on their arm, and the provider noticed a suspicious mole that should be removed. Let's say that payor only pays $10 for lesion excision. The practice can't have the patient pay for the lesion excision because $10 is insufficient reimbursement.
Hope that helps.