I tried 7mm mag with a push feed using 300WM mags, 7mm mag that came with the action and an 06 mag.
The rounds would pop out of all of the magazines.

What I would do is use a 223 bolt stroke limiter/baffle and form some sort of block in the back of the LA magazine similar to what they do with a 223 magazine on a short action.

I would try to keep the round as close to the front as possible.

You could try it with your 06 mag and see if it will feed.
The follower and mag as far as I can tell for my 7mm mag
is identical to my 06 mag but its an old action. I have a 300 WM magazine and the follower is different.

Not sure how that limiter would affect the bolt cocking either, but I would assume it should work.