Do you have USB 3 available on your computer? If so, this would be the best option, as it will allow higher burning speeds, and the USB 3 connections have sufficient power for an optical drive. If you have only USB 2, I’d stick to 4x burning.
We normally only recommend full sized external Blu-ray burners, not the slim ones. The slim drives have proved to be less robust over time, but since the manufacturers are concentrating on slim drives these days, they may be getting better. Hard to say, since we have so few people commenting on the slim ones.
In any case, I’d suggest an LG BE14NS40. It has an external power supply.
Another option would be to get a regular internal drive, like the Pioneer 209DBK and put it into a Vantec external case.
Of course with either of those options, you’ll need versions made for the UK market in order to get the proper power supplies.
Your Panasonic discs should still be fine, and are rated by users around here as some of the best on the market.