I gotta say before you get Lionheart, Axis Blade is great. By itself it's just weak, but if you equip 3 Whistlewind scarves, it makes lighting fast (sorry about the pun).
Axis blade isnt a good weapon at all in my opinion if you dont upgrade it to enkindler (ATB charge II), ATB charge doesnt make up for the super low stats. Upgrading it to enkindler requires an adamantite which you have to buy for 220000 gil. you probably wont get this much money before you activate bhakti in oerba village (which is something like 15-30min before you get lionheart). Axis blade still remains as a good choise for her, but not if you dont upgrade it to enkindler.
If you want to use the axis blade, I suggest that you stick with gladius (or blazefire saber) until you get enough money to upgrade the axis blade to enkindler. Using gladius/lionheart is still good choises.
Also, only because lionheart is named after squall (the ultimate gunblade champion) doesnt mean its Lightnings best weapon