Running both systems of balancing at the same time is where the pain begins.
In order to do so, you would need to balance your wheel with the weights, remove the valve core and then add dyna beads.
I'm sure you'll have a very decent result.
After a length of time the tire develops more inbalance than the beads can compensate for.
So you decide to have it balanced again, or add more dyna beads.
The guy at the balancing shop will scratch his head a few times, because he is continually rebalancing the tyre and each checkrun he gets a different reading.
The beads.
So he breaks the bead, sucks out the dynabeads, balances the tire and when you get home. you get your valvecore out once again and add beads to achieve perfect balance.
Great stuff if you like fiddling about.
You'd be a lot better off with one of those centramatic on board balancers.
Or either stick with the weights or the beads.
I prefer good old steel wheels and weights.
The heavier the wheel, the better balance you can obtain.
When airing up and down on a frequent basis I don't want anything inside the tire, no need to check the valvecore for leaks every time.
If your wheels and tyres start to look like a truckwheel combo, drive like a truck aka slower than usual.
If your wheels and tyres start to look tractorlike, drive like a tractor.