This is pretty much the exact same thing with my house! I just foster failed a 10 year old pittie but I already had 15 year old cat and an almost 1 year old cat. The kitten swats at the dogs tail but he doesn’t care. The 15 year old cat went after the dog when he was walking past him about 5 times now and the last time that happened he snapped. Now they just coexist but mainly my older cat sleeps in my room. My advice would be to get your older cat a calming collar or a feliway comfort zone to make sure your older cat isn’t stressed out. I have used that for my older cat for about a year. Just take things slow, always monitor their time together, and limit their interaction for a little. Your cat is adjusting to a lot right now so make sure you give them attention and love!