I’m considering informing my manager of my decision so that we can reassign some of my tasks. This could help reduce my stress, protect my mental health, and avoid complications if I start a project and then leave unexpectedly. It might also address some of the bullying, as my impending departure could reduce their leverage. I believe this approach would benefit both me and the company.
Quite the opposite. Neither party is (or should be) interested in helping the other here.
so that we can reassign some of my tasks. This could help reduce my stress, protect my mental health
Why would your employer care to reduce your workload on the promise of you leaving them soon anyway? What benefit would they get out of this? If anything, it benefits them to give you more work, as much as they can get away with, since they have lost interest in your longevity as an employee and no longer care about burning you out in the long run.
avoid complications if I start a project and then leave unexpectedly
Why should the company's difficulty in handling your absence be your problem? That is, in and of itself, a toxic workplace behavior; and you're actively enabling it by trying to behave in this way.
It's not your responsibility to manage your own staffing. That's quite literally what your employer does. It's in the name. They employ you.
It might also address some of the bullying
Your idea is to go to a bully (your own words) in a toxic workplace and give them even less of a reason to be invested in you. And you think this somehow will reduce the negative interactions and expect them to do the honorable thing?
I wouldn't even tell a good employer in advance because it unnecessarily burns bridges. Them being a bully does not make me more likely to want to freely offer information that would negatively impact me.
as my impending departure could reduce their leverage
Au contraire, it reduces your leverage. The company no longer has to consider how to keep you (which negates the cost of constantly onboarding replacements for any employee that leaves), which frees them up to make decision that no longer factor in your wellbeing, as long as they can legally get away with it.
They stop caring about burning you out or ruining your opinion of them, which gives them more freedom, not less.
I’ve started looking for a new job and I am currently interviewing with other companies. Additionally, someone in my network has mentioned that he may be able to offer me a position at his company.
Employment relies on contracts, because without a contract you have no leg to stand on when your assumption/expectation falls through.
There is no conceivable benefit from telling your current employer that you're going to leave them for someone else. Tell them when you've already signed a contract (or have a legally binding offer letter from your new employer).