Hi there. I went just went through the exact same thing with our daughter and funny enough, she is 16 months too. I would let her try it, sing, bounce, show her, everything and like you I was just so worried about it being a problem so I jumped on line and an expert said, there is a time to be super nice happy Mommy and then there is a time to be the Mother and make important things get done! And you know what, she is right. This was her advice...
After the songs and the begging, do this. Sit on the couch or chair with your child in your lap facing the same way you are. Wrap one of your legs over thier lap to prevent wiggling away, Use one of your arms across the chest to help hold their top part, or use your hand to hold their forehead so they will be still. (It sounds a little rough, but you just need to take away the wiggle room, not wrestle them to the floor, LOL)
Then take a Toddler Toothbrush, the finger kind are not as easy for you to use anymore. Stick it in their mouth, from one side of the cheek, bristle side facing out to kinda force it in there and then start brushing. The entire brushing should be for less than one minute. Not 2 whole minutes like adults.
Not only did this work for us, but I only had to do her this way for 2 nights, because after that, she knew that when I had the toothbrush, Mommy meant business and all the running and fussing was useless. =)
I hope this helps you. I know it sounds harsh, when I first read it I was like "Yeah Right Lady", but really I didn't have to be rough to hold her this way and like I said, I only had to do it for 2 oe 3 nights. So good luck cleaning those toofies. (Believe me I have a 15 yr. old and paying $5,000.00 out of pocket for Braces he won't take care of properly is no picnic either so keep thart in mind if you get discouraged. LOL