Allow for a genuine transfer day rather than treating Assisi as an effortless add-on. If dates are flexible, October 4, the Feast of St. Francis, and August 2, the Porziuncola Indulgence, are the days of greatest significance and greatest crowds.
No, not well. Assisi is about 175 km from Rome, about 2.5 hours each way by coach, so a day trip spends five hours in transit. A substantial Vatican visit alone needs 2–3 hours for the museums and Sistine Chapel plus time for St. Peter’s, and the Basilica of St. Francis, the tomb in the crypt, the Basilica of St. Clare, and the Porziuncola 4 km down in the valley need at least half a day. Choose separate days if possible; most Rome and Assisi pilgrimages give Assisi two or three nights.
If time is tight, prioritize one experience instead of treating both as brief photo stops.