Hi Gordon,
Unfortunately, I am not sure that there is a way to come up with a workaround for the TPS2114 without changing the PCB. Maybe a higher capacitance on the input would cause the voltage source to dip less and allow the current to get high enough…
Other Parts Discussed in Thread: TPS2114 Are the TPS2114 and TPS71818-33 planned to transition to pure tin lead termination finish? . TI data sheets currently specified finish as NIPDAU