Are there easy(/cheap) ways to verify this if one does buy, as an individual, a cheap battery like that? Or do you need to either rely on brand reputation or have access to expensive equipment for testing?
I think vendor reputation is almost as important (or maybe even more important) as brand reputation. In many cases a brand is just a sticker, and your vendor may well have a longer relationship with their customer.
For instance, telling a brand name cell from a fake cell can be extremely hard, but your vendor knows exactly where they bought them. I buy cells from nkon.nl, who are a high rep local vendor and I've yet to be disappointed but buying a pack that is sealed from a low rep vendor may well incur a lot of risk. You'd almost have to split open the shrink wrap to make sure you got what you paid for. One quick and dirty way to check is to weigh the pack and to compare the weight to the manufacturers spec for the cells. It's obviously not perfect but many fake cells have weights that are wildly different from the cells they pretend to copy (usually due to much lower actual capacity).
Discharge & record the curve. All it takes is one poor-performing cell & your whole pack is going to be hurting bad. Discharge down to like 2.85V * number of cells. The curve should be nice & even. If there are places where the voltage drops before the big drop at the end, it's a sign that one cell isn't performing.
If your has equalization, try to discharge & equalize charge it 2-3 times first.
If you can actually get access to the cells, measuring cell resistance is not expensive. Anomolies will show themselves very quickly. Everything should be pretty clumped together in internal-cell-resistance. If you have a pack equalizer, it can be used to help you read this out. My <$100 charger has a cell resistance test mode.
> All it takes is one poor-performing cell & your whole pack is going to be hurting bad.
Yes, that can't be stressed enough. And depending on the fault and how the BMS handles it a more dangerous condition could easily develop. For instance an undetected short would cause the other batteries to be overcharged to the point where they are much more likely to die themselves, and some of those may do so in pretty violent ways.
Cheap / missing BMS are a serious problem with cheap Lithium-Ion packs, they are the first line of defense and bad cells can even happen when supplied by good brands (but the chances are a lot higher with no-name stuff).
Another good test: charge the pack, balance it, measure the pack voltage and the individual cell voltages, then let it rest for a day or two and measure again. Any kind of deviation between cells is a sure sign something is not working well and is the earliest warning sign for trouble you can get. The rate of self discharge should be the same for all cells and it should be very low, too low to measure (and make sure you disable the balancer after the initial charge!). After a month or so it would be fine to see the pack drop a little bit, but not a lot.
Yet another quick test: use an IR camera to look at the pack both during charging and at rest. Any cells that are warmer than those around them are suspect.
You can do a capacity test quite easily, that will only tell if the capacity is the rated one. You will have to check for physical signs of bad cells, they enlarge (bloat), but that means you need to open the case (sometimes it will void any warranty) and check each cell.
