Rotted rim joist behind deck

Hello everyone. I've got an issue I need help fixing. Apologies in advance for the wall of text. When going to replace some siding I discovered that roughly 10ft of rim joist between my deck and the house is fairly rotted and needs to be replaced. The joists inside the house it is attached to are fine. It's just the rim joist that's rotted. It seems that whoever installed the deck years ago did not put any sort of flashing between the house and the deck. While this area isn't totally exposed to the elements it seems as if enough water has penetrated over the years to cause rot. The deck is supported on the far end by 4x4s and has a ledger board on the house side that is bolted to the rim joist. The deck joists are totally separate from the house joists and are not sistered. After discovering the rot I added a beam and 2 4x4 posts on the house side of the deck to support the deck. The house is a cantilevered design where the upper floor sticks out over the basement by 2ft and the bottoms of the joists are covered by a thin sheet of plywood on the outside. So I have easy access to the affected area from below. How can I go about fixing this? Obviously the best fix would be to remove the deck entirely, support everything, replace the rim joist, then rebuild. The only problem is that I don't have that sort of cash or time and the deck itself is in decent condition with no rot. Is there any way of repairing the rim joist so that everything is safe without removing the deck? Since I can access the bottom of the joists from outside is there anything that can be done from below to take care of the joist?