Will SVN allow me do this given the application_phase_2_with_sso branch?
Yes
is there any impact on the application_phase_2_with_sso branch if I kill the branch it was taken from
No. In any case you delete only in "current" state, history doens't affected, stored in repo and can can used, when needed (and, and last, due to ",...any updates to application_phase_2 have been merged-up to application_phase_2_with_sso" application_phase_2_with_sso
content is equal to "will be deleted" branch and will want nothing from parent)