.308 diameter bullets shot down a .312 bore is just plain silly.
No, it won't Kaboom. Will it shoot well, that really depends on the particular rifle and the bullet design.
I have a Pakistani built No4 MkII that seems to like .308 flat exposed lead base bullets better than it likes any factory 303 bullets of .312 diameter. It has a .310 bore.
That being said, some rifles will and some won't, just like the girls I used to date. Some did and some didn't.
Why bother though??? There are plenty of .312 diameter bullets available from 150 - 175 grain and maybe more. Most rifles won't shoot .308 bullets well.
As far as saving money???? That's a new one on me. A box of Hornady or Sierra bullets in the weights mentioned retails for the same amount as a box of .308 diameter bullets.
If you are just playing, go ahead. I doubt the jackets will obdurate enough to make a seal in the grooves but I don't think you will get any gas cutting because of it.
It's been done by thousands of people. Mostly because they just didn't know any better.
I even have a few rounds with pinch crimping made in the US during WWII for the lend lease program that are loaded with 150 grain .308 diameter bullets. Supposedly this never happened but never say never. I remember I bought several crates of them because they were cheap. Mind you, that was over 45 years ago.
The reason you are having difficulty getting your brass to size down enough to grasp 308 diameter bullets is because your expander button is sized .312. Exchange it for a .308 expander button and all will be well.