6.5 Carcano...determining proper ammo for it

I am picking up a 6.5 Carcano Cavalry tonight (11/18) and I am aware the actual round may differ a bit than what the box might say.
So, how do I measure the bore to determine what ammo would be best fit for the rifle?

remove action from stock
remove bolt from action
place barrel vertically in a bench vice with wooden blocks to prevent scaring
place cardboard box with a rag or two under muzzle
Drop a DEAD SOFT lead ball of oversized caliber into chamber (#1 buck would work well)
With a 3/16 BRASS punch,drive the shot into the throat and then fully into the rifling(if the bore is iffy,coat ball with some case lube)
push ball thru the bore and out of muzzle with cleaning rod into cardboard box
measure distance across grooves with micrometer or good calipers

A bit out of my ability to do, is this something a gunsmith can determine? (yeah, I'm new to all this, sorry)
In addition, the rifle came with some ammo, Graf and sons (by hornaday) 6.5, but when I place a round in the end of the barrel, it sticks out quite a bit, like this pic attached.
Is this normal? Or should I not fire it?

Get yourself some Cerrosafe. It’s cheap ,easy to use and will answer all of your questions. Even a caveman could do it.

Listen to Larry!

https://www.youtube.com/watch?v=KgRp3r9VPE0

Not quite sure would like to see the headstamp on the cartridge(the writing at the primer end)but that does not look like a Carcano cartridge.Do you have the box the cartridges came in? A photo of both or either would help.

They sell 6.5 but it's not cheap but can be found it's all soft top ammo, Clark brothers did carry it but if the rifle wasn't used by special forces the are generally between 125 and 250 dollars I owned one but the ammo cost and lack of stripper clips made me trade it away.

+1 for listening to Larry

Thank you, gentlemen, for your help. I had it looked at and it's been determined it's ok, and I was just unfamiliar with how the carcano was made (that .268 caliber) so hopefully I can take it out this sunday to a range somewhere and see how it does.

Cheers!

Took it out yesterday (Saturday) and it fired fine. Found some en-bloc clips from a local gun shop and they fed fine as well.
Thank you all for your kind help

6.5 Carcano ammo is easy to find. Be glad it's not 7.35