I would get a professional opinion, not someone who cuts trees. Your problem is not simple as you have two trees growing together, AND close to the house. You don't want either to come down and damage the house,

We had a maple of similar growth pattern but much smaller. On side died and eventually fell. The other half is still standing and appears healthy.

I think the myth of ”unbalancing” a tree is just that, a myth. I believe you can have the one trunk removed without bothering the other in this situation

Agree with KR Nuttle.

It's well worth it to have find someone who's insured or bonded and is experienced with this kind or work.

L Clark is correct, if done properly with the right equipment, that left side can be removed no problem.

Depending on the surrounding area, it may have to be taken down in sections using a crane or pulleys, if there's a danger of the full length trunk reaching/falling on structures or other trees that could be damaged from a tree as big as that one.

A knowledgable proffesional, will know how low they can go with the dead stump without harming the live tree.

You probably end up with a flat spot to table a large pot of flowers. ;-)

Unbalanced trees is a myth, yes, but as it happens, trees that grew like this are more likely to have other problems. If you really need the shade, keep the undamaged trunk. And hire someone at least fully licensed as a tree contractor!