One day about a year ago I decided to put a tiny fantail in my 30" planted tank, then added another a couple of months later as they are social fish and shouldn't really be kept alone IMO. I plan on moving them to a bigger tank or maybe small above ground pond in the spring though (I would either bring them inside or heat+insulate the pond over winter). So I know exactly what you mean by "just making a mess"
.
No need to cool the tank over a period of weeks, just remove the heater and keep up frequent water changes (same as when adding any fish to a new tank).
Unfortunately that tank is
waaay too small however, so hopefully this is only a temporary arrangement. If not, then you should just re-home the fish. I understand you may not want to do this as it is your daughters, but it is important to teach children the responsibilities of pet ownership. I'm not ashamed to say I learnt this the hard way crying my eyes out for my dead goldfish as a wee kid after multiple attempts at keeping fantails in a 25L tank. Once I understood the requirements for keeping very large, oxygen demanding and easily hormone-stunted fish species, it was an easy decision to rehome them as a kid. Keeping them in too small a tank is just very slow torture, they may look fine for now, but I have seen the long-term health affects from keeping goldfish in too small a tank and it is not pretty.
Personally I wouldn't trust a U2 alone for messy goldfish either, an external is really needed IMO.
If the Trigon was originally meant to be it's home...why not just plant it with hardy
Anubias, Java fern etc. attached to rocks? Or even just go without plants? Seems a shame your daughters fish is going to lose it's home from bad planning.
Edit: Just seen your tank (looks really nice), it has other tropical stock in it, so really your only ethical options are to get another large tank for the goldfish, or rehome it. Your daughter should understand if you explain why and especially explain that the fish's welfare is the most important thing
.