Hi!
It sounds like you have a prop there and your LO is using you as a soother to fall back to sleep. Would you like to change that?
It's up to you if you tackle everything at once or deal first with day time or night time. From my experience and I think there are researches about it as well, babies deal with day time sleep and night time sleep different so you won't be confusing him.
The routine in itself looks good, A times are good, length of naps are good.
IIWY, but that's just me, I would deal with the day time sleep and props first and then go on to dealing with the night. Sleep training can be hard and if you do both that means you are not getting any rest for a few days.
And who knows maybe he will get the hang of it real quick and then the nights feeds will space themselves as well. He might not be able to sleep the whole night through as he is BF, but you can get longer stretches from him.
At 5 months I would still use Shush-pat to do sleep training but some parents use PU/PD all ready. IT really depends on the baby.
Either way you need to prepare yourself to have a couple of hard days, he might be OT during those days and you may want to think about help. Perhaps you can do it over the weekend so your partner can help as well? Or your mom/friend can come and help?
You can of course do it, but if there is someone else to do it instead it might be quicker as he can smell your milk and expects to be fed.
There is info here about Shush-pat:
Shush-pat - How toand here about PU/PD:
Pick Up/Put Down (PU/PD) - Everything you ever needed to know!Good luck!