Ragdolls are notorious for spraying when they are upset ... it's not their fault, it's just how they are made. They are anxiety prone, like many of us! So, I would say - as a Mad Cat Woman who has taken care of cats for 40 years - cat 1 is distressed because (a) of the presence of cat 2 and (b) because cat 1 thinks their place at the top of the tree has been taken and 'Mummy doesn't love them any more!'
Without even meeting cat 1, I can tell you now, if you put them in the laundry room they will be so out of their minds with grief and anxiety, you'll never have clean laundry again. Please don't do it - for any cat, let alone a ragdoll, it would be cruel as, by now, they are used to being in the house and part of the family, with everything that comes with that.
Please, do not consider euthanasia. Many homes would welcome a ragdoll.
So, what to do? Try Feliway, but best of luck with that. Try a cat psychologist, but best of luck with that, the situation is probably too far gone.
For what it's worth, I think cat 1 should stay and cat 2 should be passed on to a super-duper, warm and cosy, extra-loving home. And cat 1 should be allowed to life out their little life as the only cat in the family. Betcha, soon as they are top cat again, all spraying and peeing will cease and there be a happy family once more.