The bromide may well react, but the only way to know is to try it ;-) . If you use potassium or sodium carbonate in methanol, or phase transfer conditions with hydroxide it may leave the bromide alone and has a better chance than heating with a strong base like hydroxide. The other option is to use sodium borohydride in ethanol, which might be better, but still has some risk of removing the bromide. Have a look in Greene and Wuts 'Protective groups in Organic Synthesis'

I would try it out with base as was suggested previously. If this does not work, because of the elimination, then maybe you might use a Cbz protection of the amine and remove the protecting group by hydrogenation. Good luck.

Trufluoroacetyl group may be eleminated by basic reagents better them acidic ones. You have to use minimum nucleophlisity of they. Sodium or potassiun carbonates is the best (as it was say in asnwer №1) but i'd like to add. If you want to minimise the solubility of the reagent try to use not alcoholic suspension but dioxane or even THF. Tryfluoroacetyl  group even in weakly alkaline  conditions becomes "on the nozzle hangs", and can be removed with a minimum concentration of alkaline reagent. See me work  Russian Journal of Organic Chemistry 36 (2000)(8):1173-1178 · August 2000 

Another option would be to do an amide exchange.  You could use an excess of another amine like methyl or dimethyl amine (possibly even ammonium hydroxide) and some heat to trans-amidate the trifluoroacetyl group.  Depending on the bp of the amine you chose, it may be necessary to use a pressure vessel or use a higher boiling amine like butyl amine.  

Thank you for all the answers

It is difficult to answer this type of questions without knowing the exact structure. Tertiary aliphatic bromides are not very reactive, but in strong basic conditions elimination, rather than substitution, could be a problem.