Whether a succession lawyer will give you a copy of the will while your mother is still alive will depend on a number of factors, but the main one I'm seeing is the purpose for which you require it. There's no automatic right to a copy while she's alive, because by definition it only has legal effect after she's died. Provided the POA is active (ie either authorised to commence immediately, or proof that she's lost capacity), and hasn't been revoked etc., certainly in Queensland the POA has to show that it's necessary for the POA to have a copy. POA Act s 81(1). (I imagine other states have similar provisions.)
Circumstances that might meet the test of necessity would be if you're intending to dispose of assets that you suspect may be the subject of specific gifts in the will, or significant rearrangements of her financial affairs.