Can I drill a 5/8" (16mm) hole in concrete with a cordless hammer drill?
I am not sure why they would tell you that it may ruin your drill. It might not be the most powerful hammer drill but it should be up for the job if you take the time to do it in stages.

If you cannot rent and you do not want to buy a more robust hammer drill you’re only left with one option.

I would start with a 3/16 inch masonry bit and see how that goes. Then move up to 1/4 inch or 3/8 then to 1/2 inch and finally 5/8. 

It will be easier for the drill to start with a small drill bit, once you have that hole cleared then successfully moving up in size will be less stress on the drill. If it struggles with any of those stages then you may have to do smaller incremental stages.