Hi H. - I too am unsure how to answer your question... but both my boys have AR and were on thickened formula. Both boys are gaggers and I clean up a lot of "spilling" as we call vomiting at our house.
At 10 months, your daughter must be eating some food along with her formula and juice? How is she on that? My oldest had to go to a speach therapist to learn how to eat solid foods - after about 2 1/2 months of weekly therapy he did much better. But at almost 3 years of age, he still sometimes gags and spills.
My youngest, also 10 months, has the gag refelx too. It seems to be most prevalent when it is a food he dislikes, rather than a food with a strange texture. So I have stop giving him the foods that seem to trigger the spilling.
My advice is to try to add less and less formula to each bottle until she learns to like it without. Then when you introduce milk, only use a sippy, not a bottle. Keep giving her other liquids in the sippy so she learns to drink from that as well. I don't do juice with my baby, I think I waited until my older child was closer to 2, but that is my personal choice. I do give my baby water in the sippy at 10 months.
The only other thing I would add is that the gagging seemed to get less and less as my son got older.
Good Luck.