Building up a beam using thinner and shorter material (including plywood) in the basement sound's like the best way. Don't just nail everything togeather,use yellow glue generously. BTW,don't allow BT nor HL get involved,but by all means include DT and KR.

I think I need to hire a structural engineer to inspect this, but am
thinking ahead to how we'd get a new beam into a basement, if that's the
'solution'. Anyone had to do this?

That's not your problem ... that's what the contractor who specializes in fixing house supports does. Not the OLD contractor who caused the problem, the NEW contractor.

There are various ways they can do it, depending on the space they have to work in and what's under the beam.

My BIL did this quite a bit as part of jacking up old houses, earthquake retrofitting, etc. To someone with the right experience, with the right equipment, it's not hard. It might take time, especially if they are jacking up the beam to fix settling - you can't do that all at once - but if it takes a year to slowly un-settle the house, then fix the beam, most of the year is just waiting and make a few turns to the house jacks ever month or so.

Pictures please. A crack in a beam may be nothing, especially with no symptoms.