You could try that, putting a card over the bullet. It would help, but likely would not eliminate the 'problem' if it isn't tight enough. You would have to test for accuracy of that first and second shot, no matter what you try. BP is more forgiving of small changes, than smokeless in modern guns, though.

I've not shot bullets in a SxS ML, so cannot speak from experience on the use of bullets in them.

Questions come to mind that may help identify potential solutions and problems.

Are the bullets you use engraving on the rifling when loaded or are they a loose fit?

What calibre is the gun? What are your powder charges?

Have you loaded both barrels, firing one, then reloading it and firing twice and finding the 'other' barrel's slug moved forward?

Perhaps it is tight enough to not move. I would still be concerned about this though. It would only have to happen once.

Friend of mine had a .50 caliber production rifle, that had an oversized bore for the commercial (mould) bullets he wanted to shoot. When walking with the muzzle pointed down, the bullet, loaded in a clean barrel would 'work' it's way down the bore and just barrel hang up at the muzzle with 1/2 the bullet sticking out. Once spotted, he found an easy 'fix'.

He merely wrapped the "first" bullet with a cigarette paper. That held it in the bore due to increasing the dia. slightly. Once the bore was fouled with that one shot, the bullets were tighter due to the fouling & would not 'work' forward. This was a single shot rifle, though.

Could you possibly wrap the bullet in onion-skin or cigarette paper, or perhaps bump it up so one or more bearing bands engraved to fit tighter in the bore?

Bullet movement is a serious problem. I would experiment with various means to keep the bullet on the powder.

Wad over, patched, expanded and engraving when loaded.

An engraving in the rifling when loaded (mechanical fitted) bullet should stay on the powder. If bumped hard enough by the rod after seating on the powder, it should obturate enough to hold steady- but - needs to be tested, carefully.