S.H.
He may be getting frustrated.... nursing from breast is harder, than drinking from a bottle. Nursing from breast requires mouth & muscular & tongue coordination to "suck" and get the letdown for the milk. Drinking from bottle, is easier and only requires gravity for the milk to flow.
That is a double edged sword, when breastfeeding and using bottles at the same time. Sometimes, after a time, they will just prefer a bottle...it is easier and the milk comes out easier. My son was that way.... I was breastfeeding him 95 percent of the time, but he was such a voracious feeder, that per Pediatrician, I needed to supplement him with formula. After a time, he weaned himself from breastfeeding and only wanted the bottle. That's it.
Your baby is only 11-weeks old, so yes, still feed him on demand, whenever he is hungry, even if this means waking up in the middle of the night.
ALSO though, at 12-weeks babies go through a "growth spurt"...and at these times, they feed more, more often, and more quantity. It's normal developmental based phases. (These growth spurts will occur about every 3 months or so, so head's up).
Another tip is.... if giving him a bottle supplement...do this only AFTER you have breastfed him. Usually, giving a bottle first is done to "wean" a baby off breast.
It could also be that he is fussing at the breast because there is not enough milk coming out, or the flow is not sufficient. If you offer him the bottle, don't do this to "replace"a nursing...because it can affect your milk production and lessen it.
*Perhaps see a Lactation Consultant to really evaluate your baby's latching on, and your milk flow etc.
Just some ideas, and what I have learned with my 2 children which I breastfed.
All the best,
~Susan