"Does anyone have experience with using steel beams to support sagging joists?" Yes. They're a great way to accomplish load bearing in a small package vs. wood. Your GC MAY have all the experience and expertise required to accomplish your minimally described situation.... But, I wouldn't support the beams without additional footers (as pointed out by Brickeyee) without seeing the existing structure, which may permit it. I assume you mean rectangular tube, two webs vs. one for I or H beams. Unless the former has price advantage, I'd use the latter. Leaving aside AHJ and engineering issues, why doesn't the GC address the bathroom floor alone?

Taking off an inch is a lot. For a 2x10 you lose 20% of the strength capacity and 37% of the deflection capacity when you do that. Is the floor above sagging? Seems like you should see it up there too. If not, jacking up seems out of the question. If you plane down the bottom and strengthen by adding a 2x8 to BOTH sides of the joist (glue and bolt) you should get plenty of strength and resistance to deflection out of that arrangement. Mush easier to add 2x8's because you don't have to worry about getting it flush at the top.

Two years back we did a barrier free shower in a Historical Home here in Vancouver. The floor joists where 2"x10" full dimension Fir. I needed to cut back 3" for my floor prep and the engineer would not let me. The span If I remember right was close to 18'. The fix there was sistering all the old joists with 8" LVL's (I was not allowed to install 7"). These can be ordered in 20' easy. Are you building a room above? Is there blocking courses in place?

All structural members will deflect or flex under load. That's just basic physics, not an attempt by your builder to cheat you or cause you harm or which demands remediation. The amount of deflection or flex depends on the load applied, the span, and the properties of the structural member(s). The building code specifies the maximum allowable deflection for a structural member (e.g., rafter, steel lintel, header, floor joist, ceiling joist, etc.) The amount of code-allowable deflection varies depending on the application. For example, a steel lintel supporting a masonry wall would not be allowed to deflect as much as a framed wall with a flexible cladding like vinyl siding. That should square with the "common sense" test or what engineers like to say is "intuitively obvious." For simply supported spans, the maximum deflection is in the center of the span. A ceiling joist with drywall applied is allowed to deflect l/240 where l is the length of the span. A floor is only allowed to deflect l/360, so a floor with a 12 ft. span would have a maximum allowable deflection of 0.4" That's easy to measure for a static situation. When you walk on a particular floor, you can induce a dynamic response (e.g., a bounce) which you may find objectionable and others may or may not. It's subjective. At least one manufacturer of wood I-joists conducted empirical studies of various floor assemblies and recorded whether people walking on them found them to be acceptable or not. Interestingly, there were no floor assemblies tested where 100% of people said they were acceptable. Based on their studies, their design software is able to design engineered floor systems which will be deemed acceptable at various levels. While we don't know if the OP's floor is comprised of dimensional lumber, wood I-joists or lightweight steel joists, the designer's task is the same: to meet the code requirements while balancing performance and cost. Take a look at floor structure and see if it's built as designed. If it is and the performance of the floor system is objectionable, go back to the designer and explore options to stiffen the floor.