In the current heatwave, I'd be surprised if your "coldwater" tank was not somewhere close to 24C, not too far off the temp of your "tropical" tank. Some of my tanks have peaked at ~27C in the last few days, fine for the likes of my African Butterfly and Leopard Bushfishes, but not so good for my temperates such as Ilyodon xantusi and Chaetostoma!
As for acclimitising the BN, I'm sure he will be fine. The best way would be to net him into something like a 15l bucket, ~quarter filled with his current tank water, then use some airline (with a tap or a loose knot) to drip water from the new tank into the bucket at ~5 drops per sec until the bucket fills. This will probably take the best part of an hour.
However, if the temps are similar and the tank makeup is similar (in terms of things that could alter the water chemistry like bogwood), you could simply net him straight over. But this method is more stressful on the fish, drip acclimitising is far better practice.
