Yes, this kind of trap will probably be killed by three-body losses when you scale up the density of the antimatter. Time-wise the limit is how good you can get your vacuum. This could actually work better with antimatter than with regular matter once have a macroscopic amount, because the antimatter would act as an extra vacuum pump.
Yes, storing antiprotons is not going to work beyond a few particles. I just wanted to point out that the term ionized antihydrogen is not going to be useful for anything, because if you go to the trouble of making antihydrogen (which is much harder than going for antiprotons), it would be quite stupid to ionize it again.
The most promising approach might be storing antimatter as a plasma as tried in magnetic confinement fusion research. That way you could make use of strong magnetic forces without the problems that come with a net charge of the system. Whether a trap for antimatter can be weight effective is unclear and you are right to be skeptical about that. But there is no fundamental physical reason why this cannot work and the problems are mostly technical.
Efficient production of antimatter is the much bigger problem in my opinion. The physical mechanism we use to produce antimatter is extremely inefficient so if we want to produce a macroscopic amount of antimatter we would need to find another mechanism. And we have no idea whether there is such a mechanism (well, theoretically I could propose one, but that can probably never be realized).