Individual bottle testing doesn't come into it - not for the first 3 years, at least. What it comes down to is certification for a particular design of bottle. The manufacturer has to send a batch of prototypes to the certification facility who drop them, dent them, drill holes in them and do other unpleasant things to them to verify that they meet the criteria for whatever standard they are intended to meet.
This certification process costs a lot of money, and applies to all bottles the company then makes of that exact design. If your bottle is DOT only, then the manufacturer didn't spend the extra money getting it HSE certified (or most likely it wasn't relevant for the product's intended market) and there's no way you can get it certified.
If the bottle is dual-certified, then that means that the manufacturer went through the process twice, which is one reason these bottles often cost more. Unfortunately when these bottles are retested under one system, they become technically condemned under the other, because retesting stresses the bottle outside its normal operating limits, and because the two systems have different retesting methods that differ in procedure.
Neither DOT, nor HSE, nor CE bottles are any 'safer' or 'better' than any of the others, but each country has its own safety regulations and will generally insist that those are met, rather than a foreign country's equivalent.
In the UK, ONLY bottles with a valid HSE or CE certificate are legal. In the rest of Europe, the law should require a CE label, but many countries (at least informally) allow other types. In the US they don't seem overly bothered, although you should only be using DOT-certified bottles.