In this paper, we propose a new methodology to deal with an uncertain functional input in inversion problems through computer experiments. This study is motivated by an automotive application. In this context, the simulator code takes a double set of simulation inputs : deterministic control variables and functional random variables. This framework is characterized by two features. The first feature is the high computational cost of simulations, which makes the inversion in the presence of uncertainties unaffordable. The second feature is that the probability density of the functional input V is only known through a sample of realizations. The proposed method involves two imbricated tasks. A first task based on a bayesian approach aims at wisely choosing the new evaluations of the code in order to estimate the excursion set with a limited number of costly simulations. The second task targets on efficiently estimating the expectation over the functional random variable. As the uncertain variable is observable through a finite training sample, we present three ways to infer the distribution from data. Our method is illustrated and calibrated on an analytical example. It is then applied on the automotive industrial test case where the objective is to identify the set of control parameters leading to meet the pollutant emissions standards of a vehicle.