Astrid: Gabby used to do a similar thing. She would only fall asleep when on the breast or cuddled. After fighting it for a while, I gave in and decided that if that's what works, then that's the way I was going to do it. It was a relief not to have to stress about what she was "meant" to be doing. So I just stuck to what worked to get her to sleep. She would only sleep for 30-45 minutes at a time as well. She would sleep out in the bouncer in the lounge room during the day and only sleep in her room at night.
Slowly but surely, her patterns changed. One day, when she was around 4 months old, I decided to try and get her to sleep in her room. I would put her down once a day in her cradle. She wouldn't sleep for long tho (and only after I had cuddled her to sleep).
One day I had people over and I put her down in her cradle to sleep and she settled herself!! I was shocked and rapt at the same time.
So I tried it for the next nap and the next. Before I knew it, I had a baby that settled herself and no longer wanted to be BF or cuddled to sleep. All without controlled crying! I had resigned myself to the fact that I would be cuddling her to sleep for a very long time.
Now I put her to bed awake. I do have to go in and pop the dummy in a few times but that's ok.

Anyway... while I am probably very lucky that Gabby kind of just "figured it out", my advise would be to maybe try and just go with the flow. Bubs falling asleep at the breast is not always a bad thing imo. It can be part of your routine to do it that way. Are you worried about her doing this because it's not the way it's "supposed" to be done?
As for only sleeping for short spells during the day, I really believe it's something that a lot of babies do around 3-4 months - no matter what you do! Self settling is definitely a learned thing so maybe just introduce the idea slowly??