Would you have a link to an aluminium battery? Up to now I believed they don't exist because they would be bad, both as primary and secondary ones, but am willing to change my mind if a good and working aluminium battery does exist.
Did you mean: "oxidized aluminium", or "aluminium ion", or really "aluminium oxide", which dissolves uneasily?
The difference between primary and secondary battery isn't patent, except on the datasheets. Many batteries sold as primary can be recharged a few times. It's just that the operation can burst the shell, and that the battery loses its capacity after few cycles.
For standard use, I'd expect a secondary battery to accept 500 cycles to the very least, and lose less than 30% capacity after that. Typical batteries aren't much better: they lose 20-30% capacity after 1000-3000 cycles.
Your query is a bit unexpected, because if the battery is experimental, you should have access to the designers who would tell you what they designed.