Do not use fabric or pond liner of any kind. Roots of trees are no big deal. Just cut them off, pull them out and whatever is left will become organic matter. Landscape fabric was never meant to be used for any other purpose than under gravel over soil. This prevents the fines from working their way up through the larger pores of the gravel layer and you losing your gravel as it sinks into the fines. This is what landscape fabric was made for...to save your gravel. It was never meant for weeds or roots and makes no sense to even try to find other uses. Using this stuff for weeds is flat out silly...roots, big deal. Just cut the roots off when you double dig your soil. If you've got clay I'd be very careful and rethink a rototiller. Makes concrete.
Dig a trench between the trees and your garden. Those roots become organic matter. Different soils through the soil profile just need to be broken up. I spent many many hours with my guys on projects with landscape fabric beneath crappy bark mulch...getting rid of it! The client spent a bit of moola to get their soils back to normalcy. First thing I will do if I find landscape fabric under mulch is get it out of there. Period.