Before I get into my answer, there's an error in your code btw - show-arrows="always" should be show-arrows instead. Now for what I've tested with Vuetify.js and their v-tab component...
Trial Result:
So I attempted a wide variety of fixed-tabs / grow / css widths, applying of v-flex / v-layouts to control tab size and align center... No success. I can't seem to consistently control how the v-tab component and their center-active prop behaves, especially when the browser size changes from xl to xs (i.e. desktop to mobile). Therefore, for what you're trying to achieve, I recommend taking a different approach, one that you'll be able to control consistently for different browser sizes.
Recommendation (if insistent on design with navigation arrows):
If that's the design you're fixed on, Vuetify will not easily allow you to achieve that. Vuetify relies on grouping of items / objects when moving into mobile view. For that design, I suggest you go with something like the code snippet below. Point to note, you'll have to code the chevron buttons (using @click) to select a different time. You can also apply transitions to animate the change of times.
<v-card flat dark>
<v-layout row wrap align-center>
<v-flex xs12 text-xs-center align-self-center>
<v-card-text>
<v-layout row align-center>
<v-flex>
<v-btn
flat
icon
dark
color="success"
>
<v-icon>chevron_left</v-icon>
</v-btn>
</v-flex>
<v-flex>
<div class="title font-weight-light">
{{ testValue }}
</div>
</v-flex>
<v-flex>
<v-btn
flat
icon
dark
color="success"
>
<v-icon>chevron_right</v-icon>
</v-btn>
</v-flex>
</v-layout>
</v-card-text>
</v-flex>
</v-layout>
</v-card>
Recommendation (if flexible on design):
If you wish to continue using standard components of Vuetify, you'll likely have to adopt their approach of using groups and perhaps explore a different component such as button groups or slide groups (scroll to bottom for centered active item).