A dog who turns away from a travel bowl may be distracted, dislike the bowl or not want a drink yet. That does not make a longer walk safe, especially in warm or humid weather. Bring fresh water, choose a cooler time and give them chances to rest in shade. If they keep declining, shorten the outing and return home rather than waiting for them to get thirsty enough.
Try the travel bowl at home first, where your dog is comfortable. Notice whether the same bowl works there and whether they drink normally after the walk. Never force water into their mouth or use a long walk to test a new container. The hot-day routine article offers ways to replace a risky walk, not a reason to extend the route without drinking.
If your dog is weak, unsteady, collapsing, vomiting or showing signs of overheating, get urgent veterinary help. If they are drinking less than usual at home too, contact your vet. The weather, the dog's health and the whole day's water intake matter more than whether they accepted one offer outdoors.
Today, take a familiar bowl and plan an easy way home. A safe short walk is better than a long one that depends on a drink your dog will not take.
Sources: Dogs Trust on water access and changes in drinking, Dogs Trust on hot-weather walk safety, VCA on signs of heat stroke.