Babies rarely spit up or vomit while on the breast.  Not never, but rarely.
This information can be used to help in a situation such as this.  The baby
will drink well at first, and then start to "nibble at the breast".  While
he is nibbling, he should be allowed to continue nibbling until he is
completely asleep and lets go of the breast.  With no intake from above, and
the stomach emptying at the same time, the chances of poor weight gain are
minimized, even if the spitting up itself may not stop completely.

As a second choice, I do not understand the idea of thicked formula.  If
this is truly the route to go (and the above has not been tried, I would
guess), then why not expressed milk thickened.  I would actually be willing
to wager that plain breastmilk in the bottle would decrease the spitting up.
If that is so, a nipple shield, an old fashioned one, might do the trick.

But first the falling asleep at the breast.  The mother can try lying down
with the baby to nurse, so that the baby and mother can sleep.  (If possible
with 3 others at home). Even if the baby spits up, he should gain weight.

Jack Newman, MD, FRCPC