I don't know about 'should', but if this is not working for you anymore, you can try a few gentle techniques to help re-settle him, rather than give him a bf. But be aware he probably goes back to sleep quicker with a bf than what he will (to begin with) trying to use other methods!

My DD had a similar thing start at around 16 mths and it nearly drove me around the bend. After a while I stopped giving her a bf overnight and replaced it with a sippy cup of water. I would still take her into bed to co-sleep and help her settle (she was still fairly cross about not having bb though, and made her feelings known). It took at least a week or so before she was used to the idea and started asking for a drink rather than bb. She rarely woke for a drink after that.

My only point would be that, while this sounds like it was quick, up until this stage she had been sleeping very well, it was a new-ish thing for her and may have been easier to divert. On the other hand, she was a bit older and quite determined! If you are thinking of giving it a try, I would suggest offering water (and maybe get DH to do it) for the 1am bf, rather than stop giving bfs for all of those you listed. If you can cut out those middle of the night jaunts you might find you are rested enough not to worry about the others.

Another thing is, how many bfs is he having during the day? It might make a difference if you give him an extra in the daytime.