Is it permissible to pray the obligatory prayer alone before the congregational prayer—knowing that ritual purity (wudu') is often invalidated while waiting for the congregation—and then join the congregation without wudu' with the intention of fulfilling the obligation?

Your individual prayer is valid if you are in a state of ritual purity (tahara). Your prayer with the congregation while you are unconcerned about your ablution being invalidated is a great error, as it is not permissible for one who has lost purity to pray. The Muslims have unanimously agreed on the prohibition of prayer for one who has lost purity. We advise you not to hasten to perform prayer individually, but rather to wait until the congregation is present. If you feel the urge to pass wind, leave the mosque, perform ablution, and return. If you catch the congregation, then good; otherwise, pray individually. Avoiding divergence of opinion is recommended.