Rich, the laminated beams are engineered to be stronger than 2x or 4x lumber of equivalent size. Understand though that they will not be cheap. However, when you say there is not enough room, why is that the case? From your picture, it appears that there are small cripple-studs above the opening. These cripple-studs can be cut or removed/replaced without too much hassle.
If your home was built in 1980, there is an excellent chance that the sheathing between the stucco and studs serve a "structural" purpose. The easiest way that I can think of to remove the existing header with the least disturbance to the exterior are:
(1) After severing the nails on the ends of the header (and any other exposed nail that you can see), pry a little room between the header and sheathing, enough to insert a sawzall. Run the sawzall across which should cut the nails. At that point, the header should come free.
or
(2) Run a circular saw (hd one like a worm drive skillsaw) with nail cutting blade under the header as close to the sheathing as possible. You may need to make several passes, each suceedingly deeper. This should sever most nails to the depth of 2.5." At that point, you can run the skill saw horizontally across the face pf the header on the exposed side. A few taps with hammer and/or pry bar should split the header to the extent that the cut portion can be removed. You can then make a few more horizontal passes with the saw and if the edges are free of nails, you can then remove the header in peices with the nails securing the sheathing only slightly resisting removal. When all is done, you can then cut off any remaining nails protruding from stud side of sheathing with sawzall.
After securing new header in place, remember to reattach exterior sheathing to new header to restore structural shear strength. Unfortunately, this involves screwing (not sheetrock screws) or nailing through stucco but this should minimize stucco repair work, especially if you predrill through stucco first so as not to crack the stucco. A little stucco patch and you should be fine. Just so you know, nailing density for shear wall is quite dense to satisfy code.
I'm sure this is a little more work than you had in mind but it is perfectly doable. Don't let it frighten you. You just have to go out and do it: slowly but surely, it gets done. Rather than figuring out a way to smash the header out of there, think of your job in terms of finding a way to cut all of the nails securing the header instead. Remember, the key to easy removal without smashing the heck out of everything is to cut the nails. Some finesse goes a long way towards making the job easier. These types of jobs are what makes the sawzall such an invaluable tool for demolition and remodels.