@rjh I wrote an article on EOS registration. From my own experience there is a cost for each registration and if you do not know the exact period and have to search through all periods then you will most likely run out of Ether (unless you have loads of it ). It is costly to do it twice so I would recommend just use one address and keep it safe.
https://steemit.com/eos/@nitego/eos-registration-deadline-approaches-tips-for-trezor-users