I think it's fine like that... If you want him to sleep through completely soon, try to (I said that so often now... LOL, I should have it saved somwhere) understimulate and be extremely boring at night-time. No light, no eye-contact, no talking, no nappy change, nothing. Just a quick feed and then back to bed awake. After a while he won't (if you are lucky) wake up anymore, because it's not "worth it" and just sleep through. Worked with our LO...
Another thing that worked for us: Give him milk from a bottle at night (best is expressed milk). Breastfeeding itself at night can be so nice for a baby that he wakes up just to have the comfort of the breast. Psychologically and developmentally there is nothing wrong with that, it's just very knackering for the mum.... DH can also take over and do some nightfeeds, that would make you feel better.
Good luck and well done getting as far as you did already!