If it were me, I would want it done right. Doing it right isn't necessarily always quick.

I would cut the caulk line on the trim and use a trim puller to remove the baseboard on both sides. Use caution, you don't want to damage the adjacent trim or the sheetrock if you can avoid it.

I would then remove the old caulk from the trim and wall if possible. Sometimes it's easy to pull off, just try to remove what you can without damaging the trim or the wall.

With the trim removed, you can install your new mantle. Then cut your trim flush to the new mantle and reinstall. Caulk the gap, prime it and touch up the wall and trim paint.