When my son was 13 months we had to stay with my mom for a while because I was on bed rest. She and her husband (he has 10 kids and over 60 grandkids) decided it was time for him to quit the bottle. I hadn't even thought of it. I was sort of lazy about it with my daughter too, she had the bottle until she was 15 months before I thought of switching. She had no problem switching.
ANYWAY, he was pushing away all his drinks and milk in the sippy, and like you, I was NOT HAPPY about him missing the calories, but I did want him to make the switch. He always liked a regular cup, and still does, but I knew I couldn't help him drink every single drink, so I wanted him to have sippies for a while too.
It didn't phase the grandparents, they simply offered the sippy cups only and didn't care if he pushed them away. They'd just put them back in the fridge and offer it again later. He got to have a bottle at night. After two days, if he was REALLY thirsty, he'd drink a little bit out of a sippy. It took him about 4 days, and he was happy to take the sippy for everything. He didn't starve! Good luck!