The International Society of Ultrasound advocates Intrapartum Ultrasound (US) Imaging in Obstetrics and Gynecology (ISUOG) to monitor labour progression through changes in fetal head position. Two reliable ultrasound-derived parameters that are used to predict outcomes of instrumental vaginal delivery are the angle of progression (AoP) and head-symphysis distance (HSD). In this work, as part of the Intrapartum Ultrasounds Grand Challenge (IUGC) 2024, we propose an automated fetal biometry measurement pipeline to reduce intra- and inter-observer variability and improve measurement reliability. Our pipeline consists of three key tasks: (i) classification of standard planes (SP) from US videos, (ii) segmentation of fetal head and pubic symphysis from the detected SPs, and (iii) computation of the AoP and HSD from the segmented regions. We perform sparse sampling to mitigate class imbalances and reduce spurious correlations in task (i), and utilize ensemble-based deep learning methods for task (i) and (ii) to enhance generalizability under different US acquisition settings. Finally, to promote robustness in task iii) with respect to the structural fidelity of measurements, we retain the largest connected components and apply ellipse fitting to the segmentations. Our solution achieved ACC: 0.9452, F1: 0.9225, AUC: 0.983, MCC: 0.8361, DSC: 0.918, HD: 19.73, ASD: 5.71, $Δ_{AoP}$: 8.90 and $Δ_{HSD}$: 14.35 across an unseen hold-out set of 4 patients and 224 US frames. The results from the proposed automated pipeline can improve the understanding of labour arrest causes and guide the development of clinical risk stratification tools for efficient and effective prenatal care.