I have 2 kids who share a room and they have both simply learned to sleep through the other one's various disruptions. They've been in the same room since birth and the little one never woke my daughter when he woke up overnight, I guess she just sleeps deeply? We did have a sound machine and we'd run the bathroom fan overnight for the white noise, so I'm sure that helped, maybe try that. Put the sound machine under/next to your older one's bed and that should help drown out the little one's cries.
Regarding the method, I would skip the hotel simply cause your 1yo is used to you rushing in overnight and to just put her in her room and shut the door with no sign of you would likely be distressing to her. I think you need to be there for at least the 1st few night to go in, soothe without picking up, then leave. It IS exhausting, you will be even more tired after those 3 (or so) nights than you are now. But she'll start to get the picture that she's not getting picked up and from there it gets easier.
My son was a model sleeper from birth, but when he was 11 months old we were away for nearly 3 weeks and during that time he mostly slept with me every night, was held for naps, etc. So we got home and he wanted nothing to do with his room or crib. Problem was from our bed he could easily get right to the stairs as there was no door to close and the doorways were too wide to use a baby gate (one he couldn't scale at any rate

), so the crib was a safety necessity. I had to do just what people are telling you above, go in, pat him, reassure him, but not pick him up. In the beginning I would sit in a chair next to his crib till he fell asleep on his own. Gradually I started leaving before he was asleep, but I would return if he noticed and cried, and stay a bit longer, but leave before he fell asleep again. I then would just go into his room, lay him back down, then stand at the doorway for a bit and leave. Like others have said, it was about 2 weeks (maybe a few days more) but then he was back to sleeping all night long without waking, or at least without waking me. Till he was 16 months anyway, when he learned to climb out of the crib.
And BTW, I know the exhaustion you are going through as our 1st child did not sleep through the night 1 night till she was 2 years, 5 months. It was maddening and exhausting, but NOTHING we tried worked. We had every book out there, followed them to a T, but no luck. And when we would leave her to cry she'd cry so hard she'd vomit. The vomiting got progressively sooner each night till we bagged that idea and tried something new, which also didn't work. We finally, at the advice of the pediatric sleep clinic, gave up and waited it out. At 2 years, 5 months, just days before my son was born, I told my daughter I'd move her bed to our room if she would stay in it all night long, go to bed on her own, and not wake anyone up. That was that. We had her in our room prior to that with no success, so I don't know what it was at that age, but I was thrilled that I would only have 1 waking overnight, not a newborn and a 2yo.