to M. it sounds like he is doing fine - when he stops after 3-5 minutes, does he cry and act like he is hungry, or like he is full/satisfied? As long as he is still wetting plenty fo diapers and pooping, and playing happily, I would still nurse on demand, babies know when they are really hungry, thirsty, unlike us adults, they have not lost touch with their bodies' naturaly cues. He may just be getting bigger/wanting to space out his feedings more ( don't we all wish fo this to happen, and then when it does, we don't know what is happening) - or he may just need a "snack" not a full meal, or he may just be thirsty, getting some of the thinner foremilk. and not reall HUNGRY for the fattier hindmilk - and later he gets that when he is nursing longer. Also, as they get bigger they get faster/better at getting the milk out, he might be getting all he needs for that meal in 5 minutes. Le thim take the lead, if he is not crying for more and is happy playing with his bib and hands, let him be!
At 6 months/4 months adjusted age, he does NOT NEED solids, feel free to try it, if you want, but at this point, it is not a necessity, and will just be for practice of swallowing, etc. He will still nurse just as much, and all him main nutrients come from the breastmilk.
At this age, they are also getting much more aware of the world around them, so any distraction can cause them to stop nursing if they are not really hungry - so a quiet darker room is better if available. They start to space out feedings more so they can have bigger blocks of time to work on learning other things - rolling over, holding up their heads, starting to sit up, etc, and just exploring their world! Eating, sleeping and pooping are not the only things they have to do anymore - places to go and people to see, MOM!! LOL!
Also, for the bottle feeding that he doesn't seem to enjoy from the babysitter, we found our nursing babies ONLY liked one type of bottle, this one: http://www.mambaby.com/mam/drinking-bottles,26960.html - it has a slightly flatter nipple, like the shape our breast nipples become intheir mouths, not a cylindrical shape. We never had issues with gas with these either! Also, maybe the babysitter can put a shirt of yours or a blanket that you have slept wih on her as she holds him to feed, so he can smell M..
I hope this helps some - I nursed 2 babies till 9 months each, one with reflux, and one who refused ALL bottlefeeding till 6 months!
Jessie