Is it permissible to pray the two Rak'ahs which I vowed to Allah before the 'Asr prayer, immediately after the 'Asr prayer, knowing that performing them before 'Asr sometimes leads to missing the congregational prayer?

If what was made was merely a promise and not a vow (nadhr), then it is not obligatory to fulfill it, but it is rather desirable (mustahabb). However, if it was a vow, then it is obligatory to fulfill it, based on the saying of the Prophet (peace be upon him): "Whoever vows to obey Allah, then let him obey Him." The obligatory duty takes precedence over the vow if its time is due, and then the vow is fulfilled. In this case, expiation for an oath (kaffarat yamin) is not required, and making vows is disliked (makrooh).