I had the same problem with Albus when I brought Wally home, especially with the pooping/peeing on my bed issue. Since then, I have completely banned my bed from bunnies (Wally is not allowed on ever) just to save myself from having to do laundry 3 times a day and to keep things fair. But, as to what to do to make the transition easier, I have done several things.
I actually allow Wally (second bun) playtime first, so that Albus can stay out without a strict time limit. I make sure to lie down on the floor so he can hop on my back and lick my face. I also talk to him a bit more sweetly and give him kisses on his nose (he loves kisses). I also realized that taking him out of the neutral space or shared space to a different room where just the two of us will be helps him relax and not feel like he is being replaced, especially since he has a lot of restrictions that he long since grew out of (no more getting on the couch or bathroom rug or my bed).
I guess the biggest thing is that, when Wally is out, it is a lot of bonding with me and exploring his surroundings. Wally is very curious so there is a lot of exploration. With Albus, though, it is an hour of wanting to be pet and kissed. I now do homework or watch the TV on the floor so he can cuddle with me. Listen for that teeth-grinding purr they do; I always make sure that the first session of pets always leads to Albus “purring” and being content.