Tomo the only way i could see you getting a lead free expanding bullet would be though swaging them and dies and presses to do that are very expensive. Your next best choice would be to literally glue a composite of heavy pieces of something like bismuth together in a thin jacket* and i can not see getting consistency to shoot accurately. The big problem is you are going after a big deer and i did not see what range you need to shoot at either. You need a tough bullet possibility using tin in a large caliber like 45-70 with a flat tipped bullet letting a good solid hit do the job without expansion.
. Good luck with your search i do think the Barnes bullets may for now be your safest bet .
* this would not expand it would release many pieces at impact like a shotgun at very close range while effective you need large parts to be effective.