It works a smidge better if you can find some square wire (I picked up some from riogrande (
https://www.riogrande.com/product/Copper-Square-Wire-1-Lb-Spool-14-Ga-Dead-Soft/132414) but other jewelry supply stores and probably other places.. will have it.
Otherwise it can be worth hammering it square some. If you do hammer it, annealing it will soften it back up enough to make it easier to press into the groove. Annealing can be worth it sometimes anyway for reclaimed wire if it's been wiggled around a bunch and work hardened.
To anneal heat it to a temper heat and quench in water. You can tell the temper heat by rubbing it with a sharpie and when the sharpie fades it's about close to warm enough. Ideally you'd do it in a dark room and watch for an orange glow but that's more subjective.. the sharpie is a wee bit on the cool side but close enough for most purposes. That will cause some surface corrosion so a quick wipe with a white abrasive pad (commonly sold as a "stainless steel cleaning pad") will take that right off before glue up.
If you want to harden it some after inlaying to toughen it up some lightly tapping it all over with a soft hammer will do it. You just kind of need to go back and forth over it a bunch, not even enough to deform it even, just tap tap tap will jiggle it into a harder state.