The Lapua Scenar-L .308 Winchester 175 grain Scenar-L Open Tip Match ammo will work in a Sig Cross chambered in .308. You could use this for hunting but it is not ideal. Here is a guide that could help you in selecting hunting ammo: https://www.opticsplanet.com/howto/how-to-selecting-ammo-by-application.html
I also would recommend this Nosler .308 Winchester Ballistic Tip 150 grain ammo for hunting medium game.
https://www.opticsplanet.com/nosler-308-winchester-ballistic-tip-150-grain-brass-cased-rifle-ammunition.html
Many Elk have been taken with the old Swedish 6.5×55 . 308 win. The list goes on. I have taken elk with a few different rounds I like the 300 win mag with 180grain barns X bullets. My son took one last season at 460 yards with the 6.5 creedmoor but he is a long range competition shooter. I don't recommend anything less than a 30-06 with heavy loads. Always be aware of your abilities for Elk hunting and the distance you think you will be shooting and practice. I wish you a successful Hunt.
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