yea she is gonna have to fight for it regardless. by saying she has the better claim it doesnt mean its gonna just be handed to her. whoever sits on the throne by the end of this story is gonna sit there by conquest and nothing else.
as far as who i would back tho, i would back dany. not only does she have dragons, but her ancestors made that seat to begin with and she still exists. so if i was a lord of some lands in westeros and i had an army of 5,000 and both stannis and dany asked me for support, i would take everything into consideration and back her.
at the end of the day, both have solid claims. im not saying stannis' claim is completely false. if dany didnt exist, it would undisputably be stannis. but she's out there and i would have a tough time backing stannis and his cult like army of red god worshippers. burning people at the stake for not converting religions
i dont think dany has even mention religion once. as an atheist, thats another point for her