If I brush my teeth during fasting for my A1c test, will it give a false reading?

No worries. A1c does not require fasting. That's the beauty of the test. It's not affected by what you eat before the test.
HbA1c & toothbrushin. The hemoglobin A1c is a measure of glycation found in blood tests. It is a combination of sugars meshing with proteins on the cell surface. It determines how your body processed sugar over a three month period. It is not a measure of sugar itself. Therefore, it will not be affected by teeth brushing.
Not at all. Not at all. The a1c measures blood sugar over a period of time, not the agents in your toothpaste. The flavoring agents in toothpaste are not sugar in any case. As a side comment, you could brush as effectively with no toothpaste, with mouth rinse or with baking soda and water.