Strictly speaking, you cannot avoid time dilation, but I assume what you mean is that if you took a gravbubble from Earth to a nearby star system some 11 light year away, spent a year exploring that system and came back, you'd be about 23 years older, just like the folks at home. This severely limits the usefulness of your fast gravbubble drive unless you use sleeper ships or humans are nigh-immortal and have high boredom thresholds. I wouldn't want to spent years in a cramped spaceship with no outside contact, only to be an old man when we reached our mission objective 40 light years away. This is not necessarily a bad thing, I just wanted to make sure you were aware that avoiding relativity effects in this way hampers rather than encourages exploration beyond the nearest stars. Probably not going to find a habitable world this way (and only very maybe one that can be made habitable with awesome future tech) unless you use one of the outs I mentioned. (Nor does there appear to be any reason why being in a gravbubble should influence relativistic effects.)

Your solar system is starting to look pretty amazing.