The joists run front to back not across so that's a good thing.
Well.....
Reminds me of a job I did for a woman during the pandemic. She wanted a toilet upstairs in one of the bedrooms. She was told 'sorry, not possible', and I tended to agree - initially. We had a good look at it, and without going into details [bit difficult to describe], I had to remove and refit a double fitted wardrobe, then ran a soil pipe [luckily], above ground but made a step into the bathroom, then dropped the soil pipe into the joists parallel to the joist, hit a wall, then dropped the soil pipe into the lounge [sounds nice, eh?] really tight to the ceiling and wall, and joined the stack at the back of the house. Boxed it in, and hey presto. Took ages. Looks and lots of fiddling stuff. Forget regs now, but something like 1 in 40 drop for solids and 1 in 100 for fluids, and distances? not sure..
And was she happy? she was well chuffed. And to be honest I was pleased with what I'd done too.
So, make sure you investigate the remote possibility of running a 4inch soil pipe internally first. Often it's not going to happen, but sometimes these things are possible. Sometimes it just needs somebody who is prepared to give it some thought.