If you're truly allergic to rabbits, you should rehome Bundt Cake too, not just Bear. Constant exposure to known allergens apparently makes allergies worse. Bear may have more fur, but Bundt Cake is the same species & will therefore cause problems too, just perhaps more slowly.
Are there any rabbit rescues in your area that could take Bear & Bundt Cake? The humane society here doesn't split bonded buns, nor do reputable rescues.
People who love animals are often tempted to think that only they would give x animal a good life. It's rarely true.
As for your cleaning schedule, have you tried daily? I do my buns' boxes daily. I know a lot of rabbit people do them weekly.
Also, has an allergist told you it's a rabbit allergy? If not, you're just guessing - it could be hay, straw, litter, etc.
You mention Bear's a Flemish giant. My first bun was a flemmie cross & one of my current two is a flemmie. I find that they urinate a LOT. Now both mine had/have vet diagnosed e-cuniculi. Why is this important? It affects kidneys. Both output & somehow sugar in their urine cyclically. Why does that matter? It moulds. Fast. I wonder if what you're really allergic to is microscopic mould spores. You wouldn't even notice if you were cleaning regularly, but it's there.
Before giving up a much loved pet, you owe it to all of you to investigate & pin down the true source of the allergen.
Lastly, if they're not already spayed/ neutered, you could try that. IME, intact animals cause more allergies in susceptible people than altered ones. No idea why.