Agh! I wrote you a lovely long reply and it got eaten by gremlins. So here's a shorter version!
Is PUPD the only thing that will work?
Not at all. In this case, I think Gradual Withdrawal will work best. Have a look here, but just the second half, I'm not suggesting WI/WO for you.
Walk In/Walk Out vs. The Gradual Withdrawal Method (HOW TO CHOOSE)You want LO2 going into the cot awake and happy to fall asleep there on his own. It'll take time to get there, but if you're to happy do it by taking tiny steps it might go more easily (in terms of crying).
I would start by giving him a lovely big cuddle and telling him he's going into his cot (awake). In he goes. He will protest of course. Tell him you're staying. Sit (on the floor if you have to, I always did) so he can see that you're going nowhere. Tell him it's time to sleep. Encourage him to lie himself down, if he doesn't have a lovey then now would be a good time to encourage him to adopt one. If he does, use it!
You say he's annoyed by you patting his back, so don't do it. Pat the mattress or place your hand gently on his back (without patting), speak softly to him about how it's time to sleep. Keep everything minimal. Don't worry about staying there til he's asleep at first. The next step is to work on leaving earlier.
Doing this in the evening might just help them both settle down for a better night's sleep.
In terms of the 2 yo NWs, you might want to post on the sleeping for toddler board, but 2.5-3 is prime time for bed time shenanigans so you're not alone there.
You might want to start this approach over the weekend or when you have someone else about to mind LO1 during the day. I would go with this for naps and night-time from when you start, and hopefully you will see some improvement quickly. Make sure whatever position you adopt it comfortable for you. Put cushions on the floor, or a blanket, or a chair, no point in suffering.
And if he's teething, of course, he's entitled to moan. There's no substitute for well-timed pain relief.
HTH!