Now that you you've started PUPD I would suggest sticking with it.
I will mention though, as he is soothed by rocking, when you are putting your hand on him it is possible to do a little rock motion which may help.
PUPD does use a firm hand (as you are doing) or patting (which older LOs may be frustrated by) as well as the actual picking up and putting down. If you can sooth with a firm hand or a firm hand and a little rocking *in the cot* this is a step closer to your goal. LO being soothed in the cot, feeling the weight of his body on the mattress, being alone in his cot (not in your bed) is the goal. What I'm saying is, if a little rocking helps him then I wouldn't worry about it. Rocking in the cot (just move your firm hand a little) is similar to the patting rhythm, it is also similar as you can reduce reduce reduce. Each time you use it (just as with patting) you see if you can settle in the cot, this might be more rocky (like more patting, firm) but as he settles down you can reduce it.
The steps towards achieving the end goal might be:
falling to sleep being slightly rocked in the cot
falling to sleep with a firm hand (rocking stops just before sleep)
falling to sleep with a very light hand
falling to sleep without a hand (but hand hovers only 1cm above and is ready to be replaced if needed)
falling to sleep with no hand (but you may still be in position by the cot)
falling to sleep with you out of the room
Obviously if he gets worked up you PU, then PD.
sleeping on a mattress on his floor is a great idea, he will know you are there, he may well still kick up a huge fuss but so long as you are there and responding to him he is ok. If you have a hand on him for quite a while this is also ok, he knows you are supporting him with this change.
The only thing for you is to maintain a balance (which is hard when you're exhausted!) that it is OK and right to support your baby and at the same time that you are not going to do this every night, so don't get too comfortable on that mattress because other wise he will also learn that that's where you sleep. A balance.
Other suggestion - if you have not yet introduced a key phrase this is a good time. Rather than shush, you repeat your phrase like a mantra, he hears your voice and knows you are there, it is comforting and later on this key phrase can be used from outside the room to verbally reassure. something like "Everything is ok, it's sleepy time, go to sleep" or just "go to sleep".
As you've done 3 nights now I would think you are going to get some results very soon. Often the first 3 nights are the worst. There can be a regression after a couple of good nights which is like him saying "are you really sure I'm not coming into your bed?" Fingers crossed you have a good few days now and the regression at the weekend when you have help!!
Good luck.