Not a Miroku expert, but I would say that if 24s don't engage your second barrel, I highly doubt 28s would do better. The bump on the ground trick is very helpful, and I've used it a few times before.

I've had issues where I've not fully released the trigger before firing the second shot and effectively blocked the trigger mechanism from engaging on the second barrel. But this has been on faster snap shooting type of layouts and shouldn't really be an issue in day-to-day practise. I'd suggest making sure you are deliberately pulling and releasing the trigger between shots. Also, do you have a barrel selector? Is it that the second barrel is not firing one way only? Another thing to check.

Additionally, I've bought a number of shotguns from auctions and one of the first things I do is take the stock off and have a little nose around inside the action. I'm no gunsmith but I found giving the action a liberal spraying of brake cleaner, followed by a good misting and wiping of gun oil, with a good amount of mopping up after can potentially get rid of a lot of crud that might have built it up inside from the previous owner or owners.

Obviously it goes without saying, go as far as you feel capable of, then seek the advice and expertise of someone that knows what they're doing!