I have a customer who has been on blood thinners for some time. He only has to knock his skin on something a little harder than normal and he bleeds everywhere. I get called out several times a month to remove the dried blood, normally when he comes out of hospital. Don't think he's going to be with us much longer tbh. It's a real shame as both he and his wife are really good people.
However, back to your problem. Your suggestion re: soaking with cold water is the right way to go. Let it dwell for a few mins while you set-up your extraction machine etc. You will see it start to soften almost immediately. Rinse extract with cold water, don't just vac it. Any spots left over treat with cold water again. There may be a few spots left which should shift with rust remover. Do not be tempted to use hot water at any point. M Power spotter is also good on blood. Forget treating with milk, that's just the same as blood but without the red corpuscles and enriched with calcium.
It's also handy if you have a good UV light. It will highlight any spots you may have missed.