that site says not to use goldfish primarily because feeder goldfish are frequently diseased and also cannot be later integrated into a tropical fish tank.
your goldfish aren't sick, are they? and you have a new home just waiting on them, right? then the site's reasons not to use goldfish are invalid.
as for the fishless versus with-fish cycling debate:
a big reason that it is better to perform a fishless cycle is that fish suffer long term damage from exposure to high levels of ammonia/nitrate as seen in a cycling aquarium. however, these dangerously high levels of waste chemicals are also present in severely overstocked tanks such as your unflitered 5g bowl. thus your goldfish will suffer no additional harm due to cycling and are even likely to benefit from the "vacation".
a fishless cycle might be a bit faster if it is 100% perfectly performed. but that means exhaustive daily testing to monitor waste levels, making certain not to add too much or too little ammonia, and basically draining the tank at the end of the cycle. in your case, you'll also suffer a major bacterial die-off once the birchir is introduced because your tank will be significantly understocked for the first few weeks (fishless cycling develops the maximum amount of bacteria that a system can sustain).
fishless cycling definitely has practical and ethical benefits, but when you're cramped for time, aren't familiar with the process, and still don't have a test kit--it probably isn't your best choice. (pure ammonia is also difficult to find and acquire in the UK.) cycling with fish, however, is relatively easy and mostly just involves daily 10-20% water changes. you will still want a test kit to monitor which stage of the process you're in, but ultimately the number of necessary tests will be far less.
whichever method you decide to use, just be certain to add seed bacteria from your current bowls. the addition of seed bacteria will greatly speed the cycling process.
