Well, that's about it. We haven't talked much about any point loads from bearing walls above or if the new beams have places in the walls for support. Where are your door openings? But the person doing the calculations for you can check all that. You may need stronger headers if beams fall above them.
The choice is made by getting someone to size (Engineer) the different options and price the materials and compare how much work each would be.
Redoing or replacing the existing beam will involve opening up one part of the ceiling and supporting the floor above on each side with temporary walls. Adding two additional beams will involve doing that in three places but you can do them one at a time and reuse locations of the support walls. In other words, do the existing beam first, with temporary walls on each side, then move one temporary wall to the other side to do the new beam on that side, then move them both to the other side to do the third one.
Decide if you will do the work or contract it. Then the decision makes itself based on money and trouble.
Just a footnote. If you just cut off the bottom few inches of that existing laminated beam it probably wouldn't fail, or even deflect (Sag) a whole lot more. You do know that all beams deflect, right? Even steel. But the additional deflection might cause cracks in the drywall, especially over time, and if any live load (People and stuff) was put in that room above. Residential structures are overbuilt to the point that flexing and cracks in drywall won't happen as easily. Deflections are kept to a lesser amount for framing that is covered in drywall. And you can do calculations to see of a shallower beam will support a particular load with different deflection amounts. Typical are 1/180 or 1/240 or 1/360th of span. 1 360th of a 10 foot span would be 1/3 of an inch. Beams are built with a camber built in to compensate for part of these deflections.
But these things are covered by standards so everyone knows what to expect. You can expect floors to be designed for 50#/SF LL. If you remove part of the beam and cover it up, the next guy won't know and may overload it and suffer at least a lot of cracking and a flex in the floor that doesn't feel right. So the right thing is to do it right. In fact, you are required to pull a permit to do anything structural.
My choice, and probably the least cost and trouble, if it can be done in the desired space and support the original design load, would to to rework the existing beam. Probably some combination or choice of a flitch plate beam by adding steel plates or C channels on one or both sides, or the use of additional wood beams like LVL's. Again, the calculations tell you which alternative is cheaper and easier.
Do whatever solves the problem the cheapest and easiest way.
Bill