Letting them stay and keep all their vast wealth would risk allowing them continued influence.
Also, the birth of the United Republic should involve a peaceful transfer of power, so doing this in the French style is completely out of the question.
I’m talking about what legal process is used, and what grounds are given for that process.
Of course, it would need a majority in both Houses, and in practical terms would require popular approval too.
What are their grounds for the power they currently wield??
In this case...Constitutional.
The grounds for the power they currently wield is that it’s the law.
Taking away their political powers / their status as royals is one thing, but exiling someone is punishment. You can't just hand out punishment for no good reason.
If they commited crimes that were illegal at the time they were commited they should stand trial for that, but I very much doubt any of those are serious enough to warrant exile.
While you can take away their powers pretty easily via a law everything else, such as striping their wealth, would be punishment that needs to be justified. The Queen hasn't started any wars or genocides as far as I am aware, so how would you justify this punishment?
Her being a royal is completely legal after all!