Prophet SAW and Sahaba used siwak (Mishwak) to clean their teeth and they even used it while fasting. there was no toothbrush and toothpaste at that time. so we will have to look opinion of different scholars on this issue.
according to most of scholars its completely fine to use toothbrush and paste during fast as long as you don't swallow the paste. so you can do it without any problem. Shaykh Ibn Baaz R.a has also said that its permissible.
the people who oppose usage of brush and paste during fast only promote rigidity which is against our Deen.
Lets look at the opinion of famous Muslim Scholar Dr Zakir Naik on this issue: