That sounds very difficult. He's still in that crazy newborn phase where they confuse day and night etc so that's probably part of it and will pass soon, as you probably know from your older child (my ds2 reverse cycled at that age and was just awake for ages at night, thinking it was daytime ... And once he'd sorted day from night he still only slept for half an hour at a time during the day until 6 months, which was a shock after my amazing napping ds1.).
He may also be overtired and in a cycle of being unable to sleep. Will he sleep in a sling or carrier? You won't get a rest but at least he can catch up and get less overtired, meaning you'll hopefully have a better chance of getting him off to sleep in his own bed.
Also, he probably needs his naps very close together, eg half hour nap, feed, v brief awake time, back to sleep, repeat, and if he's missing that sleep window then that may be creating the overtired cycle.
Is your older child at home all day with you? That can sometimes make it harder for the baby to sleep if they're not naturally a big sleeper to begin with. Not much you can do for that other than ride it out, I'd imagine.
Have you mentioned his discomfort to your hv or gp? If it's reflux then ranitadine or another medication may help.
Can your partner take over for several hours in the evening and morning before work to allow you to sleep a bit in the short term?