That may not preclude the possibility of screening being impacted by a lapse in appropriations, though. TSA awards the contracts for contracted screeners under the Screening Partnership Program (SPP) and I assume uses appropriated funds for most or all of the contract costs. Hopefully TSA contracting folks award the contracts such that their annual periods of performance are offset from the fiscal-year cycle to help avoid the usual new-fiscal-year shutdown risks. Yet at some point in the year each contract will need to have the next year's screening services option funded. If the time to fund that next period happens to fall during a lapse in TSA appropriations, then the contractor operating under that particular contract would cease providing services until additional funding is appropriated. And private-sector contract employees can't be forced to work without pay.
Presumably the contracts for the airports involved in the SPP have annual service option periods that begin and end on different dates, so it's unlikely that every such airport would be affected simultaneously. But the SPP isn't a guarantee against all shutdown-related impacts.
Nor would it help the air traffic control staffing issues which are more likely to be the straw that breaks the camel's back, than passenger screening slowness.