In this paper, graph attention network (GAT) is firstly utilized for the channel estimation. In accordance with the 6G expectations, we consider a high-altitude platform station (HAPS) mounted reconfigurable intelligent surface-assisted two-way communications and obtain a low overhead and a high normalized mean square error performance. The performance of the proposed method is investigated on the two-way backhauling link over the RIS-integrated HAPS. The simulation results denote that the GAT estimator overperforms the least square in full-duplex channel estimation. Contrary to the previously introduced methods, GAT at one of the nodes can separately estimate the cascaded channel coefficients. Thus, there is no need to use time-division duplex mode during pilot signaling in full-duplex communication. Moreover, it is shown that the GAT estimator is robust to hardware imperfections and changes in small-scale fading characteristics even if the training data do not include all these variations.