Pay for it, get on with it. These are the risks with property ownership.
FWIW a friend of mine wanted to build on a section which had been idle much longer than the surrounding sections. They discovered the neighbour's sewer went through the part of the section where they wanted to put the house.
So yepp, they paid to have the pipes re-routed to a non-built part of the section.
Not saying you have no legal recourse, but FFS some times you just have to move on with your life and make shi-ite happen. As others said, you will be living next door to these people.....
I think you would have to look at the different costs involved with different options. I think some basic legal advice is a good idea.
Could the OP hire a digger or dig the trench by hand to reroute it, and do most of the hard work themselves, and then just get a plumber in to do the actual plumbing stuff. Then the easement probably needs to be put on the title, and who pays for that. The laws don't seem to benefit the land owner whose land the pipe is on, when they are actually providing the benefit.
I regard to the friend above... it's a great example of the importance of obtaining a LIM which should include a plan showing services. As a neighbour had a drain traversing the property (possibly protected by a centreline or area easement, and certainly subject to section 461 LGA 1974) and located in an area where building is likely then the purchase price should have reflected this.