Rifle's with a fat chamber.

My 7400 is a little sloppy to. Fired brass out of my 7400 I measurer 0.456” diameter at the shoulder and 0.469” at the base. My S2 308 measures 0.454” diameter at the shoulder and 0.469” at the base.
Yes the body will size down like a FL. My 7400 I body size every other firing. If I was loading your rifle I would Lee Collet Die until a fire case wouldn’t chamber then body size it.
(y) I might try it. Really haven't messed with the rifle much at all yet, but it seems to shoot well so this is mostly looking down the road to prolong brass life. A positive side affect of that method being that I'm sure I've discovered there is an area that is oversized, is likely increased accuracy and easier to maintain it.
Using a small base die on accident?
Just kidding...

In all seriousness though before you try and reinvent the wheel let's look at a few things.

Both dies are doing the same thing.
Maybe you need better case lube?
Your response
"Not likely the case lube as I've done thousand's of others using it without issue"

Now maybe you're just saying of your other calibers, when reloading, your lube works fine.

One thing I learned over the years is nothing is automatic and absolutely positively don't rule out the simplest of things.

I've had rifles that felt like I didn't need hardly any lube. Matter of fact I got away with a few and no lube.
I've had others that needed the inside of the neck lubed plus a shot of spray into the die itself.

The more rifles you own or mess with little quirks pop up here and there. It's bound to happen.
I own several AR's but one was the most accurate shooting thing you'd ever see. The problem was it would cycle fine after a fresh cleaning for a about 10-15 rds. Then it wouldn't cycle. Bolt wouldn't put the next round all the way into battery.
Mag problem? No other AR has a problem (10 others)
Bolt carrier group not sliding freely like the others? It isnt. Conpany sends new bolt and carrier.
Smoother! No change though.
Take BCG from another AR that is ultra slick. Same thing. Shoots 10 rounds after a soaked cleaning...then no cycle.

Every other AR works with my loaded FL sized ammo.

Small base die? That can't help as all the other AR's work fine with regular FL die.

After constantly trying to convince myself it was many other things I dropped a FL sized empty case into the other AR chambers and made note how far gravity allowes the case to enter the chamber. Also turning them over the case just fell out.

Then did same on the problem child.

Whoa...it's not going in far enough and when I turn it over it's not falling out.

Small base die?
oh I guess...
We'll give it a go.

Yes yes yes...what do ya know...all chamber like a dream. No more hanging up.

I'm not suggesting you use a SB die....
The point is we can be our own worst enemy and often times it's the simplest of things that can cause or remedy the problem.

I've loaded thousands upon thousands of rounds and the best thing I ever did was remind myself I haven't seen everything yet and never will.
Don't overlook the simplest of simple things.

I don't know what or how you're lubing your cases but maybe try a different lubing approach especially the necks.

Spray lube and definitely hitting the outside and inside of the necks.
ESPECIALLY the necks..inside and out.
Spray them...wait a minute...
Start sizing and see what happens.