It is obvious that some folks have no idea what you are facing with the Himalayan BB, presuming that is indeed what you have.
Even pictures do not do them justice.
Yes trying to save and working around the trellis are going to run the costs up and do so significantly but the number you have been quoted is not unreasonable for a commercial cutter. Not knowing what equipment they were planning on using for your area but I know folks that quote roughly $1500 a day for that type of work and up. On Whidbey my equipment dealer says that some guys are getting $3000 a day. Of course we are not talking about a little brushhog.
If they, the BB's, are as established as I interpret from your post finding the trellis and not taking them out/damaging them will really slow the work down. A very good chance that the vines will rip them down as they are pulled/cut as you proceed. Are you certain that the trellis are not already compromised from age and the vegetative overgrowth?
Years ago I did a lot of BB work with a 6 ft Woods BrushBull and 60 HP tractor and it is slow methodical work. The Fecon mulcher/masticator is much nicer.
If you have the funds and not the time pay someone. Otherwise as some have suggest go out and give it a try. Time your work and do a measured amount of area and see what you have. May go better than you think maybe worse.
2.5 acres of heavily infested BB as I in-vision your place will be hard on your cutter and will bog it down easily, so go slow and don't let the RPM drop.
You will only know by trying.
Oh yes the seeds from the Himalayan BB can last for 5-6 years in the soil so you will need to work the re-sprouts for several years before they are all gone but it will be easier once you get the old stuff out. Even after the cutting, the roots will try again to vigorously put out new growth. Chemicals will work great. If you want to avoid that, then after they have expended the root energy to put out good growth a re-cutting before they can start to store energy back in the roots works as well. It is not and will not be a one time battle however.
Good luck
edit
Went back and noticed the 60" spacing on the trellis lines in one of your follow-on posts (post #3). Your tractor has a 61.5" wheel base according to tractordata.com specs.
You may have a problem there fitting the wheel base between the trellis rows.
None of my equipment would fit between 60" rows.