want to remove loadbearing wall
You state you got an "estimate". An estimate is not a firm price, it is not a bid, it is simply an estimate. If you want a firm price, you need a "bid", which in engineering lingo may be called a quotation. Or a contract offer. As you find the estimate extremely high, you should consult with other engineers in the area.
As a personal note, I generally avoid working for homeowners, as many of them feel the same way you do, i.e. my prices are too high. So allow me a little space here on the subject of engineering prices. I have a professional engineer's license, and am a certified cost engineer. I have been doing this for 30+ years. When I do structural engineering jobs for insurance companies, I charge $125 per hour, and that is pretty standard in our industry.
If I applied the same standard rate to a homeowner job, lets look at the time to design a header. There is a site visit, probably at least 1.5 hours travel there and back. Yes, I charge for my travel time. There is the on site consultation with the homeowner, where we discuss alternatives, such as steel vs. conventional lumber vs. laminated versus flitch beam etc. Figure an hour to discuss options, including support alternatives. I have to sketch up all the framing, which may require opening a ceiling or a floor. Figure at least an hour or two for that. I have to size the header, and draw the connection details. Figure 2-3 hours for that. Usually I have to stamp a drawing for submittal to the building department. I always discuss the project with the contractor to make sure the contractor understands all of the details, and to remind the contractor about the need for proper temporary support. I always offer to meet the contractor on site the first day to verify that the construction meets the design.
All told, this can come to 10 hours easily, which makes $1250 at my standard rate, plus mileage (yes, I charge for mileage). The majority of homeowners feel this is excessive, hence I RARELY work for homeowners, else I do it at a reduced rate when I have no other work available. So do I think the rate you were quoted is outrageous? Not if the engineer is offering a similar scope to what I generally offer. However, there may well be structural engineers in your area who are short on work, and will take the project at a discount to their normal rate. Getting a good deal on design work is not immoral, illegal or fattening, it is prudent on your part, so I suggest you shop around a bit. You may even find someone on an internet chat group willing to size the beam for you. Miracles do occur.