I'm presuming you're using wagtail-localize ...
You can add or remove fields from the translatable set using the translatable_fields list of the model/page. For you, you could force your choice field to be translatable with:
translatable_fields = translatable_fields + [
TranslatableField('some_choice_field'),
]
BUT this probably won't give you what you're hoping for since the input field for that on translation page will just be a simple input field without choice widget.
I'm not sure if it's a translation of the option chosen on the source page that you want, or if it's the ability to choose an entirely different option.
If it's a translation, then you can use a custom translation model that takes a token ID and a locale then returns text for that locale (this is how I handle static text in templates for my sites).
If you want to be able to select different options per local, you might be better using a snippet model to store your options which you can filter based on locale.