There Is no Trade-off
In the example described, the trade-off under consideration was the desire to increase national security against the freedom to walk around without an ID card in your possession at all times. There was a clear trade-off between civil freedom and improved security.
However, in the instance of a burka ban, there is no trade-off. Women that previously wore the burka are free to wear the chador, niqab or hijab. They would not lose any of their religious freedom.
Also, the fact that so few women wear the burka means that any sacrifices are very small. If banning the burka provides even a modest increase to public safety, it should be employed because the disruption would only affect a tiny proportion of the population.
On the other hand, there is potentially much to be gained in the areas of gender equality and national security.