I think it’s ok to give twigs from a tree that has codling moths. It’s the common “apple worm”. All apple trees here are affected by this moth to some extent. The larvae burr into the fruit and that’s why you should never eat apples in the dark =D

I look over the twigs I give and remove leaves that don’t look good. My bun gets apple twigs daily and this moth is common here. I give him young twigs, this- and last-year’s growth, not the older ones that can have lichen on them.

Here’s sth you could do to help fight these moths without poison, found this on Wikipedia:
“Another method for control and sampling, ‘trunk banding’, consists of wrapping a corrugated cardboard strip around the tree trunk. Larvae making their way down the tree to pupate after exiting the infested fruits will use bands as pupation sites. Bands may then be removed and burned.”